FindStat - the search engine for combinatorial statistics and maps¶
The interface to the FindStat database is
sage: findstat()
The Combinatorial Statistic Finder (https://www.findstat.org/)
>>> from sage.all import *
>>> findstat()
The Combinatorial Statistic Finder (https://www.findstat.org/)
We use the following three notions
A combinatorial collection is a set \(S\) with interesting combinatorial properties,
a combinatorial map is a combinatorially interesting map \(f: S \to S'\) between combinatorial collections, and
a combinatorial statistic is a combinatorially interesting map \(s: S \to \ZZ\).
You can use the FindStat interface to
identify a combinatorial statistic or map given the values on a few small objects,
obtain more terms, formulae, references, etc. for a given statistic or map,
edit statistics and maps and submit new statistics.
The main entry points to the database are
findstat()
to search for matching statistics,findmap()
to search for matching maps.
A guided tour¶
Retrieving information¶
The most straightforward application of the FindStat interface is to
gather information about a combinatorial statistic. To do this, we
supply findstat()
with a list of (object, value)
pairs.
For example:
sage: PM = PerfectMatchings
sage: r = findstat([(m, m.number_of_nestings()) for n in range(6) for m in PM(2*n)], depth=1); r # optional -- internet
0: St000042oMp00116 (quality [100, 100])
1: St000041 (quality [20, 100])
...
>>> from sage.all import *
>>> PM = PerfectMatchings
>>> r = findstat([(m, m.number_of_nestings()) for n in range(Integer(6)) for m in PM(Integer(2)*n)], depth=Integer(1)); r # optional -- internet
0: St000042oMp00116 (quality [100, 100])
1: St000041 (quality [20, 100])
...
The result of this query is a list (presented as a
sage.databases.oeis.FancyTuple
) of matches. Each match
consists of a FindStatCompoundStatistic
\(s: S \to \ZZ\) and
an indication of the quality of the match.
The precise meaning of the result is as follows:
The composition \(f_n \circ ... \circ f_2 \circ f_1\) applied to the objects sent to FindStat agrees with all
(object, value)
pairs of \(s\) in the database. The optional parameterdepth=1
limits the output to \(n=1\).Suppose that the quality of the match is \((q_a, q_d)\). Then \(q_a\) is the percentage of
(object, value)
pairs that are in the database among those which were sent to FindStat, and \(q_d\) is the percentage of(object, value)
pairs with distinct values in the database among those which were sent to FindStat.
Put differently, if quality
is not too small it is likely that
the statistic sent to FindStat equals \(s \circ f_n \circ ... \circ
f_2 \circ f_1\). If \(q_a\) is large, but \(q_b\) is small, then there
were many matches, but while the sought for statistic attains many
distinct values, the match found by FindStat covers only (object,
value)
pairs for few values.
In the case at hand, for the match St000041
, the list of maps is
empty. We can retrieve the description of the statistic from the
database as follows:
sage: print(r[1].statistic().description()) # optional -- internet
The number of nestings of a perfect matching.
This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$...
>>> from sage.all import *
>>> print(r[Integer(1)].statistic().description()) # optional -- internet
The number of nestings of a perfect matching.
<BLANKLINE>
<BLANKLINE>
This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$...
We can check the references:
sage: r[1].statistic().references() # optional -- internet
0: [1] de Médicis, A., Viennot, X. G., Moments des $q$-polynômes de Laguerre et la bijection de Foata-Zeilberger [[MathSciNet:1288802]]
1: [2] Simion, R., Stanton, D., Octabasic Laguerre polynomials and permutation statistics [[MathSciNet:1418763]]...
>>> from sage.all import *
>>> r[Integer(1)].statistic().references() # optional -- internet
0: [1] de Médicis, A., Viennot, X. G., Moments des $q$-polynômes de Laguerre et la bijection de Foata-Zeilberger [[MathSciNet:1288802]]
1: [2] Simion, R., Stanton, D., Octabasic Laguerre polynomials and permutation statistics [[MathSciNet:1418763]]...
If you prefer, you can look at this information also in your browser:
sage: r[1].statistic().browse() # optional -- webbrowser
>>> from sage.all import *
>>> r[Integer(1)].statistic().browse() # optional -- webbrowser
Another interesting possibility is to look for equidistributed
statistics. Instead of submitting a list of (object, value)
pairs, we pass a list of pairs (objects, values)
:
sage: data = [(PM(2*n), [m.number_of_nestings() for m in PM(2*n)]) for n in range(5)]
sage: findstat(data, depth=0) # optional -- internet
0: St000041 (quality [99, 100])
1: St000042 (quality [99, 100])
>>> from sage.all import *
>>> data = [(PM(Integer(2)*n), [m.number_of_nestings() for m in PM(Integer(2)*n)]) for n in range(Integer(5))]
>>> findstat(data, depth=Integer(0)) # optional -- internet
0: St000041 (quality [99, 100])
1: St000042 (quality [99, 100])
This results tells us that the database contains another entry that is equidistributed with the number of nestings on perfect matchings of size at most \(10\), namely the number of crossings. Note that there is a limit on the number of elements FindStat accepts for a query, which is currently \(1000\). Queries with more than \(1000\) elements are truncated.
Let us now look at a slightly more complicated example, where the submitted statistic is the composition of a sequence of combinatorial maps and a statistic known to FindStat. We use the occasion to advertise yet another way to pass values to FindStat:
sage: r = findstat(Permutations, lambda pi: pi.saliances()[0], depth=2); r # optional -- internet
0: St000740oMp00066 with offset 1 (quality [100, 100])
1: St000740oMp00087 with offset 1 (quality [100, 100])
...
>>> from sage.all import *
>>> r = findstat(Permutations, lambda pi: pi.saliances()[Integer(0)], depth=Integer(2)); r # optional -- internet
0: St000740oMp00066 with offset 1 (quality [100, 100])
1: St000740oMp00087 with offset 1 (quality [100, 100])
...
Note that some of the matches are up to a global offset. For example, we have:
sage: r[1].info() # optional -- internet
after adding 1 to every value
and applying
Mp00087: inverse first fundamental transformation: Permutations -> Permutations
to the objects (see `.compound_map()` for details)
your input matches
St000740: The last entry of a permutation.
among the values you sent, 100 percent are actually in the database,
among the distinct values you sent, 100 percent are actually in the database
>>> from sage.all import *
>>> r[Integer(1)].info() # optional -- internet
after adding 1 to every value
and applying
Mp00087: inverse first fundamental transformation: Permutations -> Permutations
to the objects (see `.compound_map()` for details)
<BLANKLINE>
your input matches
St000740: The last entry of a permutation.
<BLANKLINE>
among the values you sent, 100 percent are actually in the database,
among the distinct values you sent, 100 percent are actually in the database
Let us pick another particular result:
sage: s = findstat("St000051oMp00061oMp00069") # optional -- internet
sage: s.info() # optional -- internet
Mp00069: complement: Permutations -> Permutations
Mp00061: to increasing tree: Permutations -> Binary trees
St000051: The size of the left subtree of a binary tree.
>>> from sage.all import *
>>> s = findstat("St000051oMp00061oMp00069") # optional -- internet
>>> s.info() # optional -- internet
Mp00069: complement: Permutations -> Permutations
Mp00061: to increasing tree: Permutations -> Binary trees
St000051: The size of the left subtree of a binary tree.
To obtain the value of the statistic sent to FindStat on a given object, apply the maps in the list in the given order to this object, and evaluate the statistic on the result. For example, let us check that the result given by FindStat agrees with our statistic on the following permutation:
sage: pi = Permutation([3,1,4,5,2]); pi.saliances()[0]
3
>>> from sage.all import *
>>> pi = Permutation([Integer(3),Integer(1),Integer(4),Integer(5),Integer(2)]); pi.saliances()[Integer(0)]
3
We first have to find out, what the maps and the statistic actually do:
sage: print(s.statistic().description()) # optional -- internet
The size of the left subtree of a binary tree.
sage: print(s.statistic().sage_code()) # optional -- internet
def statistic(T):
return T[0].node_number()
sage: print("\n\n".join(m.sage_code() for m in s.compound_map())) # optional -- internet
def mapping(sigma):
return sigma.complement()
def mapping(sigma):
return sigma.increasing_tree_shape()
>>> from sage.all import *
>>> print(s.statistic().description()) # optional -- internet
The size of the left subtree of a binary tree.
>>> print(s.statistic().sage_code()) # optional -- internet
def statistic(T):
return T[0].node_number()
>>> print("\n\n".join(m.sage_code() for m in s.compound_map())) # optional -- internet
def mapping(sigma):
return sigma.complement()
<BLANKLINE>
def mapping(sigma):
return sigma.increasing_tree_shape()
So, the following should coincide with what we sent FindStat:
sage: pi.complement().increasing_tree_shape()[0].node_number()
3
>>> from sage.all import *
>>> pi.complement().increasing_tree_shape()[Integer(0)].node_number()
3
Editing and submitting statistics¶
Of course, often a statistic will not be in the database:
sage: s = findstat([(d, randint(1,1000)) for d in DyckWords(4)]); s # optional -- internet
St000000: a new statistic on Dyck paths
>>> from sage.all import *
>>> s = findstat([(d, randint(Integer(1),Integer(1000))) for d in DyckWords(Integer(4))]); s # optional -- internet
St000000: a new statistic on Dyck paths
In this case, and if the statistic might be “interesting”, please
consider submitting it to the database using
FindStatStatistic.submit()
.
Also, you may notice omissions, typos or even mistakes in the
description, the code and the references. In this case, simply
replace the value by using FindStatFunction.set_description()
,
FindStatStatistic.set_code()
or
FindStatFunction.set_references_raw()
, and then
FindStatStatistic.submit()
your changes for review by the
FindStat team.
AUTHORS:
Martin Rubey (2015): initial version
Martin Rubey (2020): rewrite, adapt to new FindStat API
- class sage.databases.findstat.FindStat[source]¶
Bases:
UniqueRepresentation
,SageObject
The Combinatorial Statistic Finder.
FindStat
is a class preserving user information.- browse()[source]¶
Open the FindStat web page in a browser.
EXAMPLES:
sage: findstat().browse() # optional -- webbrowser
>>> from sage.all import * >>> findstat().browse() # optional -- webbrowser
- login()[source]¶
Open the FindStat login page in a browser.
EXAMPLES:
sage: findstat().login() # optional -- webbrowser
>>> from sage.all import * >>> findstat().login() # optional -- webbrowser
- set_user(name=None, email=None)[source]¶
Set the user for the session.
INPUT:
name
– the name of the useremail
– an email address of the user
This information is used when submitting a statistic with
FindStatStatistic.submit()
.EXAMPLES:
sage: findstat().set_user(name='Anonymous', email='invalid@org')
>>> from sage.all import * >>> findstat().set_user(name='Anonymous', email='invalid@org')
Note
It is usually more convenient to login into the FindStat web page using the
login()
method.
- user_email()[source]¶
Return the user name used for submissions.
EXAMPLES:
sage: findstat().set_user(name='Anonymous', email='invalid@org') sage: findstat().user_email() 'invalid@org'
>>> from sage.all import * >>> findstat().set_user(name='Anonymous', email='invalid@org') >>> findstat().user_email() 'invalid@org'
- user_name()[source]¶
Return the user name used for submissions.
EXAMPLES:
sage: findstat().set_user(name='Anonymous', email='invalid@org') sage: findstat().user_name() 'Anonymous'
>>> from sage.all import * >>> findstat().set_user(name='Anonymous', email='invalid@org') >>> findstat().user_name() 'Anonymous'
- class sage.databases.findstat.FindStatCollection(parent, id, data, sageconstructor_overridden)[source]¶
Bases:
Element
A FindStat collection.
FindStatCollection
is a class representing a combinatorial collection available in the FindStat database.Its main use is to allow easy specification of the combinatorial collection when using
findstat
. It also provides methods to quickly access its FindStat web page (browse()
), check whether a particular element is actually in the range considered by FindStat (in_range()
), etc.INPUT:
One of the following:
a string eg. ‘Dyck paths’ or ‘DyckPaths’, case-insensitive, or
an integer designating the FindStat id of the collection, or
a Sage object belonging to a collection, or
an iterable producing a Sage object belonging to a collection.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Dyck paths") # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(5) # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(DyckWord([1,0,1,0])) # optional -- internet Cc0005: Dyck paths sage: FindStatCollection(DyckWords(2)) # optional -- internet a subset of Cc0005: Dyck paths sage: FindStatCollection(DyckWords) # optional -- internet Cc0005: Dyck paths
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> FindStatCollection("Dyck paths") # optional -- internet Cc0005: Dyck paths >>> FindStatCollection(Integer(5)) # optional -- internet Cc0005: Dyck paths >>> FindStatCollection(DyckWord([Integer(1),Integer(0),Integer(1),Integer(0)])) # optional -- internet Cc0005: Dyck paths >>> FindStatCollection(DyckWords(Integer(2))) # optional -- internet a subset of Cc0005: Dyck paths >>> FindStatCollection(DyckWords) # optional -- internet Cc0005: Dyck paths
See also
- browse()[source]¶
Open the FindStat web page of the collection in a browser.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Permutations").browse() # optional -- webbrowser
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> FindStatCollection("Permutations").browse() # optional -- webbrowser
- element_level(element)[source]¶
Return the level of an element.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: cc = FindStatCollection("Perfect Matchings") # optional -- internet sage: cc.element_level(PerfectMatching([[1,2],[3,4],[5,6]])) # optional -- internet 6
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> cc = FindStatCollection("Perfect Matchings") # optional -- internet >>> cc.element_level(PerfectMatching([[Integer(1),Integer(2)],[Integer(3),Integer(4)],[Integer(5),Integer(6)]])) # optional -- internet 6
- elements_on_level(level)[source]¶
Return an iterable over the elements on the given level.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Perfect Matchings").elements_on_level(4) # optional -- internet Perfect matchings of {1, 2, 3, 4}
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> FindStatCollection("Perfect Matchings").elements_on_level(Integer(4)) # optional -- internet Perfect matchings of {1, 2, 3, 4}
- first_terms(function, level=None)[source]¶
Compute the first few terms of the given function, possibly restricted to a level, as a lazy list.
INPUT:
function
– a callablelevel
– (optional), the level to restrict to
OUTPUT:
A lazy list of pairs of the form
(object, value)
.EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.first_terms(lambda x: 1)[:10].list() # optional -- internet [([[1, 0], [0]], 1), ([[1, 0], [1]], 1), ([[2, 0], [0]], 1), ([[2, 0], [1]], 1), ([[2, 0], [2]], 1), ([[1, 1], [1]], 1), ([[1, 0, 0], [0, 0], [0]], 1), ([[1, 0, 0], [1, 0], [0]], 1), ([[1, 0, 0], [1, 0], [1]], 1), ([[3, 0], [0]], 1)]
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet >>> c.first_terms(lambda x: Integer(1))[:Integer(10)].list() # optional -- internet [([[1, 0], [0]], 1), ([[1, 0], [1]], 1), ([[2, 0], [0]], 1), ([[2, 0], [1]], 1), ([[2, 0], [2]], 1), ([[1, 1], [1]], 1), ([[1, 0, 0], [0, 0], [0]], 1), ([[1, 0, 0], [1, 0], [0]], 1), ([[1, 0, 0], [1, 0], [1]], 1), ([[3, 0], [0]], 1)]
- from_string()[source]¶
Return a function that returns the object given a FindStat representation.
OUTPUT:
The function that produces the Sage object given its FindStat representation as a string.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("Posets") # optional -- internet sage: p = c.from_string()('([(0, 2), (2, 1)], 3)') # optional -- internet sage: p.cover_relations() # optional -- internet [[0, 2], [2, 1]] sage: c = FindStatCollection("Binary Words") # optional -- internet sage: w = c.from_string()('010101') # optional -- internet sage: w in c._data["Code"].elements_on_level(6) # optional -- internet True
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> c = FindStatCollection("Posets") # optional -- internet >>> p = c.from_string()('([(0, 2), (2, 1)], 3)') # optional -- internet >>> p.cover_relations() # optional -- internet [[0, 2], [2, 1]] >>> c = FindStatCollection("Binary Words") # optional -- internet >>> w = c.from_string()('010101') # optional -- internet >>> w in c._data["Code"].elements_on_level(Integer(6)) # optional -- internet True
- id()[source]¶
Return the FindStat identifier of the collection.
OUTPUT: the FindStat identifier of the collection as an integer
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.id() # optional -- internet 18
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet >>> c.id() # optional -- internet 18
- id_str()[source]¶
Return the FindStat identifier of the collection.
OUTPUT: the FindStat identifier of the collection as a string
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet sage: c.id_str() # optional -- internet 'Cc0018'
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> c = FindStatCollection("GelfandTsetlinPatterns") # optional -- internet >>> c.id_str() # optional -- internet 'Cc0018'
- in_range(element)[source]¶
Check whether an element of the collection is in FindStat’s precomputed range.
INPUT:
element
– a Sage object that belongs to the collection
OUTPUT:
True
, ifelement
is used by the FindStat search engine, andFalse
if it is ignoredEXAMPLES:
sage: # optional - internet sage: from sage.databases.findstat import FindStatCollection sage: c = FindStatCollection("GelfandTsetlinPatterns") sage: c.in_range(GelfandTsetlinPattern([[2, 1], [1]])) True sage: c.in_range(GelfandTsetlinPattern([[3, 1], [1]])) True sage: c.in_range(GelfandTsetlinPattern([[7, 1], [1]])) False
>>> from sage.all import * >>> # optional - internet >>> from sage.databases.findstat import FindStatCollection >>> c = FindStatCollection("GelfandTsetlinPatterns") >>> c.in_range(GelfandTsetlinPattern([[Integer(2), Integer(1)], [Integer(1)]])) True >>> c.in_range(GelfandTsetlinPattern([[Integer(3), Integer(1)], [Integer(1)]])) True >>> c.in_range(GelfandTsetlinPattern([[Integer(7), Integer(1)], [Integer(1)]])) False
- is_element(element)[source]¶
Return whether the element belongs to the collection.
If the collection is not yet supported, return whether element is a string.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: cc = FindStatCollection("Perfect Matchings") # optional -- internet sage: cc.is_element(PerfectMatching([[1,2],[3,4],[5,6]])) # optional -- internet True sage: cc.is_element(SetPartition([[1,2],[3,4],[5,6]])) # optional -- internet False
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> cc = FindStatCollection("Perfect Matchings") # optional -- internet >>> cc.is_element(PerfectMatching([[Integer(1),Integer(2)],[Integer(3),Integer(4)],[Integer(5),Integer(6)]])) # optional -- internet True >>> cc.is_element(SetPartition([[Integer(1),Integer(2)],[Integer(3),Integer(4)],[Integer(5),Integer(6)]])) # optional -- internet False
- is_supported()[source]¶
Check whether the collection is fully supported by the interface.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection(1).is_supported() # optional -- internet True sage: FindStatCollection(24).is_supported() # optional -- internet, random False
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> FindStatCollection(Integer(1)).is_supported() # optional -- internet True >>> FindStatCollection(Integer(24)).is_supported() # optional -- internet, random False
- levels_with_sizes()[source]¶
Return a dictionary from levels to level sizes.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: cc = FindStatCollection("Perfect Matchings") # optional -- internet sage: cc.levels_with_sizes() # optional -- internet {2: 1, 4: 3, 6: 15, 8: 105, 10: 945}
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> cc = FindStatCollection("Perfect Matchings") # optional -- internet >>> cc.levels_with_sizes() # optional -- internet {2: 1, 4: 3, 6: 15, 8: 105, 10: 945}
- name(style='singular')[source]¶
Return the name of the FindStat collection.
INPUT:
style
– string (default:'singular'
); can be'singular'
, or'plural'
OUTPUT: the name of the FindStat collection, in singular or in plural
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: FindStatCollection("Binary trees").name() # optional -- internet 'Binary tree' sage: FindStatCollection("Binary trees").name(style='plural') # optional -- internet 'Binary trees'
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> FindStatCollection("Binary trees").name() # optional -- internet 'Binary tree' >>> FindStatCollection("Binary trees").name(style='plural') # optional -- internet 'Binary trees'
- to_string()[source]¶
Return a function that returns a FindStat representation given an object.
If the collection is not yet supported, return the identity.
OUTPUT:
The function that produces the string representation as needed by the FindStat search webpage.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCollection sage: p = Poset((range(3), [[0, 1], [1, 2]])) # optional -- internet sage: c = FindStatCollection("Posets") # optional -- internet sage: c.to_string()(p) # optional -- internet '([(0, 1), (1, 2)], 3)'
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollection >>> p = Poset((range(Integer(3)), [[Integer(0), Integer(1)], [Integer(1), Integer(2)]])) # optional -- internet >>> c = FindStatCollection("Posets") # optional -- internet >>> c.to_string()(p) # optional -- internet '([(0, 1), (1, 2)], 3)'
- class sage.databases.findstat.FindStatCollections[source]¶
Bases:
UniqueRepresentation
,Parent
The class of FindStat collections.
The elements of this class are combinatorial collections in FindStat as of January 2020. If a new collection was added to the web service since then, the dictionary
_SupportedFindStatCollections
in this module has to be updated accordingly.EXAMPLES:
sage: from sage.databases.findstat import FindStatCollections sage: sorted(c for c in FindStatCollections() if c.is_supported()) # optional -- internet [Cc0001: Permutations, Cc0002: Integer partitions, Cc0005: Dyck paths, Cc0006: Integer compositions, Cc0007: Standard tableaux, Cc0009: Set partitions, Cc0010: Binary trees, Cc0012: Perfect matchings, Cc0013: Cores, Cc0014: Posets, Cc0017: Alternating sign matrices, Cc0018: Gelfand-Tsetlin patterns, Cc0019: Semistandard tableaux, Cc0020: Graphs, Cc0021: Ordered trees, Cc0022: Finite Cartan types, Cc0023: Parking functions, Cc0024: Binary words, Cc0025: Plane partitions, Cc0026: Decorated permutations, Cc0027: Signed permutations, Cc0028: Skew partitions, Cc0029: Lattices, Cc0030: Ordered set partitions]
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollections >>> sorted(c for c in FindStatCollections() if c.is_supported()) # optional -- internet [Cc0001: Permutations, Cc0002: Integer partitions, Cc0005: Dyck paths, Cc0006: Integer compositions, Cc0007: Standard tableaux, Cc0009: Set partitions, Cc0010: Binary trees, Cc0012: Perfect matchings, Cc0013: Cores, Cc0014: Posets, Cc0017: Alternating sign matrices, Cc0018: Gelfand-Tsetlin patterns, Cc0019: Semistandard tableaux, Cc0020: Graphs, Cc0021: Ordered trees, Cc0022: Finite Cartan types, Cc0023: Parking functions, Cc0024: Binary words, Cc0025: Plane partitions, Cc0026: Decorated permutations, Cc0027: Signed permutations, Cc0028: Skew partitions, Cc0029: Lattices, Cc0030: Ordered set partitions]
- Element[source]¶
alias of
FindStatCollection
- class sage.databases.findstat.FindStatCombinatorialMap[source]¶
Bases:
SageObject
A class serving as common ancestor of
FindStatStatistic
andFindStatCompoundStatistic
.
- class sage.databases.findstat.FindStatCombinatorialStatistic[source]¶
Bases:
SageObject
A class providing methods to retrieve the first terms of a statistic.
This class provides methods applicable to instances of
FindStatStatistic
,FindStatCompoundStatistic
andFindStatStatisticQuery
.- first_terms()[source]¶
Return the first terms of the (compound) statistic as a dictionary.
OUTPUT:
A dictionary from Sage objects representing an element of the appropriate collection to integers.
This method is overridden in
FindStatStatisticQuery
.EXAMPLES:
sage: findstat(41).first_terms()[PerfectMatching([(1,6),(2,5),(3,4)])] # optional -- internet 3
>>> from sage.all import * >>> findstat(Integer(41)).first_terms()[PerfectMatching([(Integer(1),Integer(6)),(Integer(2),Integer(5)),(Integer(3),Integer(4))])] # optional -- internet 3
- first_terms_str(max_values=1200)[source]¶
Return the first terms of the statistic in the format needed for a FindStat query.
OUTPUT:
A string, where each line is of the form
object => value
, whereobject
is the string representation of an element of the appropriate collection as used by FindStat and value is an integer.EXAMPLES:
sage: print(findstat(41).first_terms_str(max_values=4)) # optional -- internet [(1,2)] => 0 [(1,2),(3,4)] => 0 [(1,3),(2,4)] => 0 [(1,4),(2,3)] => 1
>>> from sage.all import * >>> print(findstat(Integer(41)).first_terms_str(max_values=Integer(4))) # optional -- internet [(1,2)] => 0 [(1,2),(3,4)] => 0 [(1,3),(2,4)] => 0 [(1,4),(2,3)] => 1
- generating_functions(style='polynomial', max_values=1200)[source]¶
Return the generating functions of the statistic as a dictionary.
The keys of this dictionary are the levels for which the generating function of the statistic can be computed from the known data. Each value represents a generating function for one level, as a polynomial, as a dictionary, or as a list of coefficients.
INPUT:
style
– string (default:'polynomial'
); can be'polynomial'
,'dictionary'
, or'list'
OUTPUT:
if
style
is'polynomial'
, the generating function is returned as a polynomialif
style
is'dictionary'
, the generating function is returned as a dictionary representing the monomials of the generating functionif
style
is'list'
, the generating function is returned as a list of coefficients of the generating function. In this case, leading and trailing zeros are omitted.
EXAMPLES:
sage: st = findstat(41) # optional -- internet sage: st.generating_functions() # optional -- internet {2: 1, 4: q + 2, 6: q^3 + 3*q^2 + 6*q + 5, 8: q^6 + 4*q^5 + 10*q^4 + 20*q^3 + 28*q^2 + 28*q + 14} sage: st.generating_functions(style='dictionary') # optional -- internet {2: {0: 1}, 4: {0: 2, 1: 1}, 6: {0: 5, 1: 6, 2: 3, 3: 1}, 8: {0: 14, 1: 28, 2: 28, 3: 20, 4: 10, 5: 4, 6: 1}} sage: st.generating_functions(style='list') # optional -- internet {2: [1], 4: [2, 1], 6: [5, 6, 3, 1], 8: [14, 28, 28, 20, 10, 4, 1]}
>>> from sage.all import * >>> st = findstat(Integer(41)) # optional -- internet >>> st.generating_functions() # optional -- internet {2: 1, 4: q + 2, 6: q^3 + 3*q^2 + 6*q + 5, 8: q^6 + 4*q^5 + 10*q^4 + 20*q^3 + 28*q^2 + 28*q + 14} >>> st.generating_functions(style='dictionary') # optional -- internet {2: {0: 1}, 4: {0: 2, 1: 1}, 6: {0: 5, 1: 6, 2: 3, 3: 1}, 8: {0: 14, 1: 28, 2: 28, 3: 20, 4: 10, 5: 4, 6: 1}} >>> st.generating_functions(style='list') # optional -- internet {2: [1], 4: [2, 1], 6: [5, 6, 3, 1], 8: [14, 28, 28, 20, 10, 4, 1]}
- oeis_search(search_size=32, verbose=True)[source]¶
Search the OEIS for the generating function of the statistic.
INPUT:
search_size
– (default: 32) the number of integers in the sequence. If this is chosen too big, the OEIS result may be corrupted.verbose
– boolean (default:True
); ifTrue
, some information about the search are printed
OUTPUT: a tuple of OEIS sequences, see
sage.databases.oeis.OEIS.find_by_description()
for more informationEXAMPLES:
sage: st = findstat(41) # optional -- internet sage: st.oeis_search() # optional -- internet Searching the OEIS for "1 2,1 5,6,3,1 14,28,28,20,10,4,1" 0: A067311: Triangle read by rows: T(n,k) gives number of ways of arranging n chords on a circle with k simple intersections ...
>>> from sage.all import * >>> st = findstat(Integer(41)) # optional -- internet >>> st.oeis_search() # optional -- internet Searching the OEIS for "1 2,1 5,6,3,1 14,28,28,20,10,4,1" 0: A067311: Triangle read by rows: T(n,k) gives number of ways of arranging n chords on a circle with k simple intersections ...
- class sage.databases.findstat.FindStatCompoundMap(id, domain=None, codomain=None, check=True)[source]¶
Bases:
Element
,FindStatCombinatorialMap
Initialize a compound statistic.
INPUT:
id
– a padded identifierdomain
– (optional), the domain of the compound mapcodomain
– (optional), the codomain of the compound mapcheck
– whether to check that domains and codomains fit
If domain and codomain are given and
check
isFalse
, they are not fetched from FindStat.If
id
is the empty string,domain
must be provided, and the identity map on this collection is returned.- browse()[source]¶
Open the FindStat web page of the compound map in a browser.
EXAMPLES:
sage: findmap(62).browse() # optional -- webbrowser
>>> from sage.all import * >>> findmap(Integer(62)).browse() # optional -- webbrowser
- codomain()[source]¶
Return the codomain of the compound map.
EXAMPLES:
sage: findmap("Mp00099oMp00127").codomain() # optional -- internet Cc0005: Dyck paths
>>> from sage.all import * >>> findmap("Mp00099oMp00127").codomain() # optional -- internet Cc0005: Dyck paths
- domain()[source]¶
Return the domain of the compound map.
EXAMPLES:
sage: findmap("Mp00099oMp00127").domain() # optional -- internet Cc0001: Permutations
>>> from sage.all import * >>> findmap("Mp00099oMp00127").domain() # optional -- internet Cc0001: Permutations
- id_str()[source]¶
Return the padded identifier of the compound map.
EXAMPLES:
sage: findmap("Mp00099oMp00127").id_str() # optional -- internet 'Mp00099oMp00127'
>>> from sage.all import * >>> findmap("Mp00099oMp00127").id_str() # optional -- internet 'Mp00099oMp00127'
- info()[source]¶
Print a detailed explanation of the compound map.
EXAMPLES:
sage: findmap("Mp00099oMp00127").info() # optional -- internet Mp00127: left-to-right-maxima to Dyck path: Permutations -> Dyck paths Mp00099: bounce path: Dyck paths -> Dyck paths
>>> from sage.all import * >>> findmap("Mp00099oMp00127").info() # optional -- internet Mp00127: left-to-right-maxima to Dyck path: Permutations -> Dyck paths Mp00099: bounce path: Dyck paths -> Dyck paths
- maps()[source]¶
Return the maps occurring in the compound map as a list.
EXAMPLES:
sage: findmap("Mp00099oMp00127").maps() # optional -- internet [Mp00127: left-to-right-maxima to Dyck path, Mp00099: bounce path]
>>> from sage.all import * >>> findmap("Mp00099oMp00127").maps() # optional -- internet [Mp00127: left-to-right-maxima to Dyck path, Mp00099: bounce path]
- class sage.databases.findstat.FindStatCompoundStatistic(id, domain=None, check=True)[source]¶
Bases:
Element
,FindStatCombinatorialStatistic
Initialize a compound statistic.
A compound statistic is a sequence of maps followed by a statistic.
INPUT:
id
– a padded identifierdomain
– (optional), the domain of the compound statisticcheck
– whether to check that domains and codomains fit
If the domain is given and
check
isFalse
, it is not fetched from FindStat.- browse()[source]¶
Open the FindStat web page of the compound statistic in a browser.
EXAMPLES:
sage: from sage.databases.findstat import FindStatCompoundStatistic sage: FindStatCompoundStatistic("St000042oMp00116").browse() # optional -- webbrowser
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCompoundStatistic >>> FindStatCompoundStatistic("St000042oMp00116").browse() # optional -- webbrowser
- compound_map()[source]¶
Return the compound map which is part of the compound statistic.
EXAMPLES:
sage: findstat("St000051oMp00061oMp00069").compound_map() # optional -- internet Mp00061oMp00069
>>> from sage.all import * >>> findstat("St000051oMp00061oMp00069").compound_map() # optional -- internet Mp00061oMp00069
- domain()[source]¶
Return the domain of the compound statistic.
EXAMPLES:
sage: findstat("St000042oMp00116").domain() # optional -- internet Cc0012: Perfect matchings
>>> from sage.all import * >>> findstat("St000042oMp00116").domain() # optional -- internet Cc0012: Perfect matchings
- id_str()[source]¶
Return the padded identifier of the compound statistic.
EXAMPLES:
sage: findstat("St000042oMp00116").id_str() # optional -- internet 'St000042oMp00116'
>>> from sage.all import * >>> findstat("St000042oMp00116").id_str() # optional -- internet 'St000042oMp00116'
- info()[source]¶
Print a detailed description of the compound statistic.
EXAMPLES:
sage: findstat("St000042oMp00116").info() # optional -- internet Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings St000042: The number of crossings of a perfect matching.
>>> from sage.all import * >>> findstat("St000042oMp00116").info() # optional -- internet Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings St000042: The number of crossings of a perfect matching.
- statistic()[source]¶
Return the statistic of the compound statistic.
EXAMPLES:
sage: findstat("St000041oMp00116").statistic() # optional -- internet St000041: The number of nestings of a perfect matching.
>>> from sage.all import * >>> findstat("St000041oMp00116").statistic() # optional -- internet St000041: The number of nestings of a perfect matching.
- class sage.databases.findstat.FindStatFunction(id, data=None, function=None)[source]¶
Bases:
SageObject
A class providing the common methods of
FindStatMap
andFindStatStatistic
.This class provides methods to access and modify properties of a single statistic or map of the FindStat database.
- description()[source]¶
Return the description of the statistic or map.
OUTPUT: string; for statistics, the first line is used as name
EXAMPLES:
sage: print(findstat(51).description()) # optional -- internet The size of the left subtree of a binary tree.
>>> from sage.all import * >>> print(findstat(Integer(51)).description()) # optional -- internet The size of the left subtree of a binary tree.
- domain()[source]¶
Return the FindStat domain of the statistic or map.
OUTPUT:
The domain of the statistic or map as an instance of
FindStatCollection
.EXAMPLES:
sage: findstat(51).domain() # optional -- internet Cc0010: Binary trees sage: findmap(62).domain() # optional -- internet Cc0001: Permutations
>>> from sage.all import * >>> findstat(Integer(51)).domain() # optional -- internet Cc0010: Binary trees >>> findmap(Integer(62)).domain() # optional -- internet Cc0001: Permutations
- id()[source]¶
Return the FindStat identifier of the statistic or map.
OUTPUT: the FindStat identifier of the statistic or map, as an integer
EXAMPLES:
sage: findstat(51).id() # optional -- internet 51
>>> from sage.all import * >>> findstat(Integer(51)).id() # optional -- internet 51
- id_str()[source]¶
Return the FindStat identifier of the statistic or map.
OUTPUT: the FindStat identifier of the statistic or map, as a string
EXAMPLES:
sage: findstat(51).id_str() # optional -- internet 'St000051'
>>> from sage.all import * >>> findstat(Integer(51)).id_str() # optional -- internet 'St000051'
- name()[source]¶
Return the name of the statistic or map.
OUTPUT:
A string. For statistics, this is just the first line of the description.
EXAMPLES:
sage: findstat(51).name() # optional -- internet 'The size of the left subtree of a binary tree.'
>>> from sage.all import * >>> findstat(Integer(51)).name() # optional -- internet 'The size of the left subtree of a binary tree.'
- references()[source]¶
Return the references associated with the statistic or map.
OUTPUT:
An instance of
sage.databases.oeis.FancyTuple
, each item corresponds to a reference.EXAMPLES:
sage: findstat(41).references() # optional -- internet 0: [1] de Médicis, A., Viennot, X. G., Moments des $q$-polynômes de Laguerre et la bijection de Foata-Zeilberger [[MathSciNet:1288802]] 1: [2] Simion, R., Stanton, D., Octabasic Laguerre polynomials and permutation statistics [[MathSciNet:1418763]]
>>> from sage.all import * >>> findstat(Integer(41)).references() # optional -- internet 0: [1] de Médicis, A., Viennot, X. G., Moments des $q$-polynômes de Laguerre et la bijection de Foata-Zeilberger [[MathSciNet:1288802]] 1: [2] Simion, R., Stanton, D., Octabasic Laguerre polynomials and permutation statistics [[MathSciNet:1418763]]
- references_raw()[source]¶
Return the unrendered references associated with the statistic or map.
EXAMPLES:
sage: print(findstat(41).references_raw()) # optional -- internet [1] [[MathSciNet:1288802]] [2] [[MathSciNet:1418763]]
>>> from sage.all import * >>> print(findstat(Integer(41)).references_raw()) # optional -- internet [1] [[MathSciNet:1288802]] [2] [[MathSciNet:1418763]]
- reset()[source]¶
Discard all modification of the statistic or map.
EXAMPLES:
sage: s = findmap(62) # optional -- internet sage: s.set_name(u"Möbius"); s # optional -- internet Mp00062(modified): Möbius sage: s.reset(); s # optional -- internet Mp00062: Lehmer-code to major-code bijection
>>> from sage.all import * >>> s = findmap(Integer(62)) # optional -- internet >>> s.set_name(u"Möbius"); s # optional -- internet Mp00062(modified): Möbius >>> s.reset(); s # optional -- internet Mp00062: Lehmer-code to major-code bijection
- sage_code()[source]¶
Return the Sage code associated with the statistic or map.
OUTPUT: an empty string or a string of the form:
def statistic(x): ...
or:
def mapping(x): ...
EXAMPLES:
sage: print(findstat(51).sage_code()) # optional -- internet def statistic(T): return T[0].node_number()
>>> from sage.all import * >>> print(findstat(Integer(51)).sage_code()) # optional -- internet def statistic(T): return T[0].node_number()
- set_description(value)[source]¶
Set the description of the statistic or map.
INPUT:
value
– string; for statistics, this is the name of the statistic followed by its description on a separate line
This information is used when submitting the statistic or map with
submit()
.EXAMPLES:
sage: q = findstat([(d, randint(1, 1000)) for d in DyckWords(4)]) # optional -- internet sage: q.set_description("Random values on Dyck paths.\nNot for submission.") # optional -- internet sage: print(q.description()) # optional -- internet Random values on Dyck paths. Not for submission.
>>> from sage.all import * >>> q = findstat([(d, randint(Integer(1), Integer(1000))) for d in DyckWords(Integer(4))]) # optional -- internet >>> q.set_description("Random values on Dyck paths.\nNot for submission.") # optional -- internet >>> print(q.description()) # optional -- internet Random values on Dyck paths. Not for submission.
- set_references_raw(value)[source]¶
Set the references associated with the statistic or map.
INPUT:
value
– string; each reference should be on a single line, and consist of one or more links to the same item
FindStat will automatically resolve the links, if possible. A complete list of supported services can be found at <https://findstat.org/NewStatistic>.
This information is used when submitting the statistic with
submit()
.EXAMPLES:
sage: q = findstat([(d, randint(1, 1000)) for d in DyckWords(4)]) # optional -- internet sage: q.set_references_raw("[[arXiv:1102.4226]]\n[[oeis:A000001]]") # optional -- internet sage: q.references() # optional -- internet 0: [[arXiv:1102.4226]] 1: [[oeis:A000001]]
>>> from sage.all import * >>> q = findstat([(d, randint(Integer(1), Integer(1000))) for d in DyckWords(Integer(4))]) # optional -- internet >>> q.set_references_raw("[[arXiv:1102.4226]]\n[[oeis:A000001]]") # optional -- internet >>> q.references() # optional -- internet 0: [[arXiv:1102.4226]] 1: [[oeis:A000001]]
- set_sage_code(value)[source]¶
Set the code associated with the statistic or map.
INPUT:
value
– string; SageMath code producing the values of the statistic or map
Contributors are encouraged to submit code for statistics using
FindStatStatistic.set_code()
. Modifying the “verified” SageMath code using this method is restricted to members of the FindStatCrew, for all other contributors this method has no effect.EXAMPLES:
sage: q = findstat([(d, randint(1,1000)) for d in DyckWords(4)]) # optional -- internet sage: q.set_sage_code("def statistic(x):\n return randint(1, 1000)") # optional -- internet sage: print(q.sage_code()) # optional -- internet def statistic(x): return randint(1,1000)
>>> from sage.all import * >>> q = findstat([(d, randint(Integer(1),Integer(1000))) for d in DyckWords(Integer(4))]) # optional -- internet >>> q.set_sage_code("def statistic(x):\n return randint(1, 1000)") # optional -- internet >>> print(q.sage_code()) # optional -- internet def statistic(x): return randint(1,1000)
- class sage.databases.findstat.FindStatMap(parent, id)[source]¶
Bases:
Element
,FindStatFunction
,FindStatCombinatorialMap
A FindStat map.
FindStatMap
is a class representing a combinatorial map available in the FindStat database.This class provides methods to inspect various properties of these maps, in particular
code()
.EXAMPLES:
sage: from sage.databases.findstat import FindStatMap sage: FindStatMap(116) # optional -- internet Mp00116: Kasraoui-Zeng
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMap >>> FindStatMap(Integer(116)) # optional -- internet Mp00116: Kasraoui-Zeng
See also
- browse()[source]¶
Open the FindStat web page of the map in a browser.
EXAMPLES:
sage: from sage.databases.findstat import FindStatMap sage: FindStatMap(116).browse() # optional -- webbrowser
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMap >>> FindStatMap(Integer(116)).browse() # optional -- webbrowser
- codomain()[source]¶
Return the FindStat collection which is the codomain of the map.
OUTPUT: the codomain of the map as a
FindStatCollection
EXAMPLES:
sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: FindStatMap(27).codomain() # optional -- internet Cc0002: Integer partitions
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMap # optional -- internet >>> FindStatMap(Integer(27)).codomain() # optional -- internet Cc0002: Integer partitions
- info()[source]¶
Print a detailed description of the map.
EXAMPLES:
sage: findmap("Mp00116").info() # optional -- internet Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings
>>> from sage.all import * >>> findmap("Mp00116").info() # optional -- internet Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings
- properties_raw()[source]¶
Return the properties of the map.
OUTPUT: the properties as a string
EXAMPLES:
sage: from sage.databases.findstat import FindStatMap # optional -- internet sage: FindStatMap(61).properties_raw() # optional -- internet 'surjective, graded'
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMap # optional -- internet >>> FindStatMap(Integer(61)).properties_raw() # optional -- internet 'surjective, graded'
- set_name(value)[source]¶
Set the name of the map.
INPUT:
value
– string; the new name of the map
This information is used when submitting the map with
submit()
.
- set_properties_raw(value)[source]¶
Set the properties of the map.
EXAMPLES:
sage: # optional - internet sage: from sage.databases.findstat import FindStatMap sage: FindStatMap(61).set_properties_raw('surjective') sage: FindStatMap(61).properties_raw() 'surjective' sage: FindStatMap(61) Mp00061(modified): to increasing tree sage: FindStatMap(61).reset() sage: FindStatMap(61) Mp00061: to increasing tree
>>> from sage.all import * >>> # optional - internet >>> from sage.databases.findstat import FindStatMap >>> FindStatMap(Integer(61)).set_properties_raw('surjective') >>> FindStatMap(Integer(61)).properties_raw() 'surjective' >>> FindStatMap(Integer(61)) Mp00061(modified): to increasing tree >>> FindStatMap(Integer(61)).reset() >>> FindStatMap(Integer(61)) Mp00061: to increasing tree
- class sage.databases.findstat.FindStatMapQuery(data=None, values_of=None, distribution_of=None, domain=None, codomain=None, known_terms=None, function=None, depth=2, debug=False)[source]¶
Bases:
FindStatMap
A class representing a query for FindStat (compound) maps.
- class sage.databases.findstat.FindStatMaps(domain=None, codomain=None)[source]¶
Bases:
UniqueRepresentation
,Parent
The class of FindStat maps.
The elements of this class are combinatorial maps currently in FindStat.
EXAMPLES:
We can print a sample map for each domain and codomain:
sage: from sage.databases.findstat import FindStatCollections, FindStatMaps sage: ccs = sorted(FindStatCollections())[:3] # optional -- internet sage: for cc_dom in ccs: # optional -- internet ....: for cc_codom in ccs: ....: print(cc_dom.name(style='plural') + " -> " + cc_codom.name(style='plural')) ....: try: ....: print(" " + next(iter(FindStatMaps(cc_dom, cc_codom))).name()) ....: except StopIteration: ....: pass Permutations -> Permutations Lehmer-code to major-code bijection Permutations -> Integer partitions Robinson-Schensted tableau shape Permutations -> Dyck paths left-to-right-maxima to Dyck path Integer partitions -> Permutations Integer partitions -> Integer partitions conjugate Integer partitions -> Dyck paths to Dyck path Dyck paths -> Permutations to non-crossing permutation Dyck paths -> Integer partitions to partition Dyck paths -> Dyck paths reverse
>>> from sage.all import * >>> from sage.databases.findstat import FindStatCollections, FindStatMaps >>> ccs = sorted(FindStatCollections())[:Integer(3)] # optional -- internet >>> for cc_dom in ccs: # optional -- internet ... for cc_codom in ccs: ... print(cc_dom.name(style='plural') + " -> " + cc_codom.name(style='plural')) ... try: ... print(" " + next(iter(FindStatMaps(cc_dom, cc_codom))).name()) ... except StopIteration: ... pass Permutations -> Permutations Lehmer-code to major-code bijection Permutations -> Integer partitions Robinson-Schensted tableau shape Permutations -> Dyck paths left-to-right-maxima to Dyck path Integer partitions -> Permutations Integer partitions -> Integer partitions conjugate Integer partitions -> Dyck paths to Dyck path Dyck paths -> Permutations to non-crossing permutation Dyck paths -> Integer partitions to partition Dyck paths -> Dyck paths reverse
- Element[source]¶
alias of
FindStatMap
- class sage.databases.findstat.FindStatMatchingMap(matching_map, quality, domain=None, codomain=None)[source]¶
Bases:
FindStatCompoundMap
Initialize a FindStat map match.
INPUT:
matching_map
– a compound map identifierquality
– the quality of the match, as provided by FindStatdomain
– (optional) the domain of the compound mapcodomain
– (optional), the codomain of the compound map
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingMap sage: FindStatMatchingMap("Mp00099oMp00127", [83]) # optional -- internet Mp00099oMp00127 (quality [83])
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingMap >>> FindStatMatchingMap("Mp00099oMp00127", [Integer(83)]) # optional -- internet Mp00099oMp00127 (quality [83])
- info()[source]¶
Print a detailed explanation of the match.
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingMap sage: FindStatMatchingMap("Mp00099oMp00127", [83]).info() # optional -- internet your input matches Mp00127: left-to-right-maxima to Dyck path: Permutations -> Dyck paths Mp00099: bounce path: Dyck paths -> Dyck paths among the values you sent, 83 percent are actually in the database
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingMap >>> FindStatMatchingMap("Mp00099oMp00127", [Integer(83)]).info() # optional -- internet your input matches Mp00127: left-to-right-maxima to Dyck path: Permutations -> Dyck paths Mp00099: bounce path: Dyck paths -> Dyck paths <BLANKLINE> among the values you sent, 83 percent are actually in the database
- quality()[source]¶
Return the quality of the match, as provided by FindStat.
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingMap sage: FindStatMatchingMap("Mp00099oMp00127", [83]).quality() # optional -- internet [83]
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingMap >>> FindStatMatchingMap("Mp00099oMp00127", [Integer(83)]).quality() # optional -- internet [83]
- class sage.databases.findstat.FindStatMatchingStatistic(matching_statistic, offset, quality, domain=None)[source]¶
Bases:
FindStatCompoundStatistic
Initialize a FindStat statistic match.
INPUT:
matching_statistic
– a compound statistic identifieroffset
– the offset of the values, as provided by FindStatquality
– the quality of the match, as provided by FindStatdomain
– (optional) the domain of the compound statistic
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingStatistic sage: FindStatMatchingStatistic("St000042oMp00116", 1, [17, 83]) # optional -- internet St000042oMp00116 with offset 1 (quality [17, 83])
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingStatistic >>> FindStatMatchingStatistic("St000042oMp00116", Integer(1), [Integer(17), Integer(83)]) # optional -- internet St000042oMp00116 with offset 1 (quality [17, 83])
- info()[source]¶
Print a detailed explanation of the match.
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingStatistic sage: r = FindStatMatchingStatistic("St000042oMp00116", 1, [17, 83]) # optional -- internet sage: r.info() # optional -- internet after adding 1 to every value and applying Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings to the objects (see `.compound_map()` for details) your input matches St000042: The number of crossings of a perfect matching. among the values you sent, 17 percent are actually in the database, among the distinct values you sent, 83 percent are actually in the database sage: r = FindStatMatchingStatistic("St000042", 1, [17, 83]) # optional -- internet sage: r.info() # optional -- internet after adding 1 to every value your input matches St000042: The number of crossings of a perfect matching. among the values you sent, 17 percent are actually in the database, among the distinct values you sent, 83 percent are actually in the database
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingStatistic >>> r = FindStatMatchingStatistic("St000042oMp00116", Integer(1), [Integer(17), Integer(83)]) # optional -- internet >>> r.info() # optional -- internet after adding 1 to every value and applying Mp00116: Kasraoui-Zeng: Perfect matchings -> Perfect matchings to the objects (see `.compound_map()` for details) <BLANKLINE> your input matches St000042: The number of crossings of a perfect matching. <BLANKLINE> among the values you sent, 17 percent are actually in the database, among the distinct values you sent, 83 percent are actually in the database >>> r = FindStatMatchingStatistic("St000042", Integer(1), [Integer(17), Integer(83)]) # optional -- internet >>> r.info() # optional -- internet after adding 1 to every value <BLANKLINE> your input matches St000042: The number of crossings of a perfect matching. <BLANKLINE> among the values you sent, 17 percent are actually in the database, among the distinct values you sent, 83 percent are actually in the database
- offset()[source]¶
Return the offset which has to be added to each value of the compound statistic to obtain the desired value.
EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingStatistic sage: r = FindStatMatchingStatistic("St000042oMp00116", 1, [17, 83]) # optional -- internet sage: r.offset() # optional -- internet 1
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingStatistic >>> r = FindStatMatchingStatistic("St000042oMp00116", Integer(1), [Integer(17), Integer(83)]) # optional -- internet >>> r.offset() # optional -- internet 1
- quality()[source]¶
Return the quality of the match, as provided by FindStat.
The quality of a statistic match is a pair of percentages \((q_a, q_d)\), where \(q_a\) is the percentage of
(object, value)
pairs that are in the database among those which were sent to FindStat, and \(q_d\) is the percentage of(object, value)
pairs with distinct values in the database among those which were sent to FindStat.EXAMPLES:
sage: from sage.databases.findstat import FindStatMatchingStatistic sage: r = FindStatMatchingStatistic("St000042oMp00116", 1, [17, 83]) # optional -- internet sage: r.quality() # optional -- internet [17, 83]
>>> from sage.all import * >>> from sage.databases.findstat import FindStatMatchingStatistic >>> r = FindStatMatchingStatistic("St000042oMp00116", Integer(1), [Integer(17), Integer(83)]) # optional -- internet >>> r.quality() # optional -- internet [17, 83]
- class sage.databases.findstat.FindStatStatistic(parent, id)[source]¶
Bases:
Element
,FindStatFunction
,FindStatCombinatorialStatistic
A FindStat statistic.
FindStatStatistic
is a class representing a combinatorial statistic available in the FindStat database.This class provides methods to inspect and update various properties of these statistics.
EXAMPLES:
sage: from sage.databases.findstat import FindStatStatistic sage: FindStatStatistic(41) # optional -- internet St000041: The number of nestings of a perfect matching.
>>> from sage.all import * >>> from sage.databases.findstat import FindStatStatistic >>> FindStatStatistic(Integer(41)) # optional -- internet St000041: The number of nestings of a perfect matching.
See also
- browse()[source]¶
Open the FindStat web page of the statistic in a browser.
EXAMPLES:
sage: findstat(41).browse() # optional -- webbrowser
>>> from sage.all import * >>> findstat(Integer(41)).browse() # optional -- webbrowser
- code()[source]¶
Return the code associated with the statistic or map.
OUTPUT: string
Contributors are encouraged to submit Sage code in the form:
def statistic(x): ...
but the string may also contain code for other computer algebra systems.
EXAMPLES:
sage: print(findstat(41).code()) # optional -- internet def statistic(x): return len(x.nestings()) sage: print(findstat(118).code()) # optional -- internet (* in Mathematica *) tree = {{{{}, {}}, {{}, {}}}, {{{}, {}}, {{}, {}}}}; Count[tree, {{___}, {{___}, {{___}, {___}}}}, {0, Infinity}]
>>> from sage.all import * >>> print(findstat(Integer(41)).code()) # optional -- internet def statistic(x): return len(x.nestings()) >>> print(findstat(Integer(118)).code()) # optional -- internet (* in Mathematica *) tree = {{{{}, {}}, {{}, {}}}, {{{}, {}}, {{}, {}}}}; Count[tree, {{___}, {{___}, {{___}, {___}}}}, {0, Infinity}]
- edit(max_values=1200)[source]¶
Open the FindStat web page for editing the statistic or submitting a new statistic in a browser.
- info()[source]¶
Print a detailed description of the statistic.
EXAMPLES:
sage: findstat("St000042").info() # optional -- internet St000042: The number of crossings of a perfect matching.
>>> from sage.all import * >>> findstat("St000042").info() # optional -- internet St000042: The number of crossings of a perfect matching.
- set_code(value)[source]¶
Set the code associated with the statistic.
INPUT:
value
– string; code producing the values of the statistic
Contributors are encouraged to submit SageMath code in the form:
def statistic(x): ...
However, code for any other platform is accepted also.
This information is used when submitting the statistic with
submit()
.EXAMPLES:
sage: q = findstat([(d, randint(1,1000)) for d in DyckWords(4)]) # optional -- internet sage: q.set_code("def statistic(x):\n return randint(1,1000)") # optional -- internet sage: print(q.code()) # optional -- internet def statistic(x): return randint(1,1000)
>>> from sage.all import * >>> q = findstat([(d, randint(Integer(1),Integer(1000))) for d in DyckWords(Integer(4))]) # optional -- internet >>> q.set_code("def statistic(x):\n return randint(1,1000)") # optional -- internet >>> print(q.code()) # optional -- internet def statistic(x): return randint(1,1000)
- set_first_terms(values)[source]¶
Update the first terms of the statistic.
INPUT:
values
– list of pairs of the form(object, value)
whereobject
is a Sage object representing an element of the appropriate collection andvalue
is an integer
This information is used when submitting the statistic with
submit()
.Warning
This method cannot check whether the given values are actually correct. Moreover, it does not even perform any sanity checks.
- class sage.databases.findstat.FindStatStatisticQuery(data=None, values_of=None, distribution_of=None, domain=None, known_terms=None, function=None, depth=2, debug=False)[source]¶
Bases:
FindStatStatistic
A class representing a query for FindStat (compound) statistics.
- first_terms(max_values=1200)[source]¶
Return the pairs of the known terms which contain singletons as a dictionary.
EXAMPLES:
sage: PM = PerfectMatchings sage: l = [(PM(2*n), [m.number_of_nestings() for m in PM(2*n)]) for n in range(5)] sage: r = findstat(l, depth=0); r # optional -- internet 0: St000041 (quality [99, 100]) 1: St000042 (quality [99, 100]) sage: r.first_terms() # optional -- internet {[]: 0, [(1, 2)]: 0}
>>> from sage.all import * >>> PM = PerfectMatchings >>> l = [(PM(Integer(2)*n), [m.number_of_nestings() for m in PM(Integer(2)*n)]) for n in range(Integer(5))] >>> r = findstat(l, depth=Integer(0)); r # optional -- internet 0: St000041 (quality [99, 100]) 1: St000042 (quality [99, 100]) >>> r.first_terms() # optional -- internet {[]: 0, [(1, 2)]: 0}
- class sage.databases.findstat.FindStatStatistics(domain=None)[source]¶
Bases:
UniqueRepresentation
,Parent
The class of FindStat statistics.
The elements of this class are combinatorial statistics currently in FindStat.
EXAMPLES:
We can print a list of the first few statistics currently in FindStat in a given domain:
sage: from sage.databases.findstat import FindStatStatistics sage: for st, _ in zip(FindStatStatistics("Perfect Matchings"), range(3)): # optional -- internet ....: print(" " + st.name()) The number of nestings of a perfect matching. The number of crossings of a perfect matching. The number of crossings plus two-nestings of a perfect matching.
>>> from sage.all import * >>> from sage.databases.findstat import FindStatStatistics >>> for st, _ in zip(FindStatStatistics("Perfect Matchings"), range(Integer(3))): # optional -- internet ... print(" " + st.name()) The number of nestings of a perfect matching. The number of crossings of a perfect matching. The number of crossings plus two-nestings of a perfect matching.
- Element[source]¶
alias of
FindStatStatistic
- sage.databases.findstat.findmap(*args, **kwargs)[source]¶
Return matching maps.
INPUT:
Valid keywords are:
domain
,codomain
,values
,distribution
,depth
andmax_values
. They have the following meanings:depth
– nonnegative integer (default:FINDSTAT_DEFAULT_DEPTH
); specifying how many maps to apply to generate the given mapmax_values
– integer (default:FINDSTAT_MAX_VALUES
); specifying how many values are sent to the finderdomain
,codomain
– integer or string of the formCc1234
, designates the domain and codomain of the sought for mapsvalues
,distribution
– data specifying the values or distribution of values of the sought for maps. The keyword argumentsdepth
andmax_values
are passed to the finder. The data may be specified in one of the following forms:a list of pairs of the form
(object, value)
, or a dictionary from Sage objects to Sage objects.a list of pairs of the form
(list of objects, list of values)
, or a single pair of the form(list of objects, list of values)
. In each pair there should be as many objects as values.a callable. In this case, the domain must be specified, also. The callable is then used to generate
max_values
(object, value)
pairs.The number of terms generated may also be controlled by passing an iterable collection, such as
Permutations(3)
.
findmap
also accepts at most three positional arguments as follows:a single positional argument, if none of
domain
,codomain
,values
ordistribution
are specified, is interpreted as a FindStat map identifier. If further arguments are given and it is a string, it is interpreted as a domain. If all this fails, it is interpreted as the specification of values.if two positional arguments are given, the first is interpreted as domain, and the second either as codomain or the specification of values.
if three positional arguments are given, the first two are interpreted as domain and codomain, and the third as the specification of values.
OUTPUT:
An instance of a
FindStatMap
,FindStatMapQuery
orFindStatMaps
.EXAMPLES:
A particular map can be retrieved by its Mp-identifier or number:
sage: findmap('Mp00062') # optional -- internet Mp00062: Lehmer-code to major-code bijection sage: findmap(62) # optional -- internet Mp00062: Lehmer-code to major-code bijection sage: findmap("Mp00099oMp00127") # optional -- internet Mp00099oMp00127
>>> from sage.all import * >>> findmap('Mp00062') # optional -- internet Mp00062: Lehmer-code to major-code bijection >>> findmap(Integer(62)) # optional -- internet Mp00062: Lehmer-code to major-code bijection >>> findmap("Mp00099oMp00127") # optional -- internet Mp00099oMp00127
The database can be searched by providing a list of pairs:
sage: l = [pi for n in range(5) for pi in Permutations(n)] sage: findmap([(pi, pi.complement().increasing_tree_shape()) for pi in l], depth=2) # optional -- internet 0: Mp00061oMp00069 (quality [...])
>>> from sage.all import * >>> l = [pi for n in range(Integer(5)) for pi in Permutations(n)] >>> findmap([(pi, pi.complement().increasing_tree_shape()) for pi in l], depth=Integer(2)) # optional -- internet 0: Mp00061oMp00069 (quality [...])
or a dictionary:
sage: findmap({pi: pi.complement().increasing_tree_shape() for pi in l}, depth=2) # optional -- internet 0: Mp00061oMp00069 (quality [...])
>>> from sage.all import * >>> findmap({pi: pi.complement().increasing_tree_shape() for pi in l}, depth=Integer(2)) # optional -- internet 0: Mp00061oMp00069 (quality [...])
Note however, that the results of these two queries need not compare equal, because we compare queries by the data sent, and the ordering of the data might be different.
Another possibility is to send a collection and a function. In this case, the function is applied to the first few objects of the collection:
sage: findmap("Permutations", lambda pi: pi.increasing_tree_shape(), depth=1) # optional -- internet 0: Mp00061 (quality [100])
>>> from sage.all import * >>> findmap("Permutations", lambda pi: pi.increasing_tree_shape(), depth=Integer(1)) # optional -- internet 0: Mp00061 (quality [100])
In rare cases, it may not be possible to guess the codomain of a map, in which case it can be provided as second argument or keyword argument:
sage: findmap("Dyck paths", "Perfect matchings", lambda D: [(a+1, b) for a,b in D.tunnels()]) # optional -- internet 0: Mp00146 (quality [100]) sage: findmap("Dyck paths", "Set partitions", lambda D: [(a+1, b) for a,b in D.tunnels()]) # optional -- internet 0: Mp00092oMp00146 (quality [...])
>>> from sage.all import * >>> findmap("Dyck paths", "Perfect matchings", lambda D: [(a+Integer(1), b) for a,b in D.tunnels()]) # optional -- internet 0: Mp00146 (quality [100]) >>> findmap("Dyck paths", "Set partitions", lambda D: [(a+Integer(1), b) for a,b in D.tunnels()]) # optional -- internet 0: Mp00092oMp00146 (quality [...])
Finally, we can also retrieve all maps with a given domain or codomain:
sage: findmap("Cc0024") # optional -- internet Set of combinatorial maps with domain Cc0024: Binary words used by FindStat sage: findmap(codomain='Cores') # optional -- internet Set of combinatorial maps with codomain Cc0013: Cores used by FindStat
>>> from sage.all import * >>> findmap("Cc0024") # optional -- internet Set of combinatorial maps with domain Cc0024: Binary words used by FindStat >>> findmap(codomain='Cores') # optional -- internet Set of combinatorial maps with codomain Cc0013: Cores used by FindStat
- sage.databases.findstat.findstat(query=None, values=None, distribution=None, domain=None, depth=2, max_values=1000)[source]¶
Return matching statistics.
INPUT:
One of the following:
an integer or a string representing a valid FindStat identifier (e.g. 45 or ‘St000045’). The keyword arguments
depth
andmax_values
are ignored,values
anddistribution
must beNone
.a list of pairs of the form
(object, value)
, or a dictionary from Sage objects to integer values. The keyword argumentsdepth
andmax_values
are passed to the finder,values
anddistribution
must beNone
.a list of pairs of the form (list of objects, list of values), or a single pair of the form (list of objects, list of values). In each pair there should be as many objects as values. The keyword arguments
depth
andmax_values
are passed to the finder.a collection and a list of pairs of the form (string, value), or a dictionary from strings to integer values. The keyword arguments
depth
andmax_values
are passed to the finder. This should only be used if the collection is not yet supported.a collection and a callable. The callable is used to generate
max_values
(object, value)
pairs. The number of terms generated may also be controlled by passing an iterable collection, such asPermutations(3)
. The keyword argumentsdepth
andmax_values
are passed to the finder.
OUTPUT: an instance of a
FindStatStatistic
, represented bythe FindStat identifier together with its name, or
a list of triples, each consisting of
the statistic
a list of strings naming certain maps
a number which says how many of the values submitted agree with the values in the database, when applying the maps in the given order to the object and then computing the statistic on the result.
EXAMPLES:
A particular statistic can be retrieved by its St-identifier or number:
sage: findstat('St000041') # optional -- internet St000041: The number of nestings of a perfect matching. sage: findstat(51) # optional -- internet St000051: The size of the left subtree of a binary tree. sage: findstat('St000042oMp00116') # optional -- internet St000042oMp00116
>>> from sage.all import * >>> findstat('St000041') # optional -- internet St000041: The number of nestings of a perfect matching. >>> findstat(Integer(51)) # optional -- internet St000051: The size of the left subtree of a binary tree. >>> findstat('St000042oMp00116') # optional -- internet St000042oMp00116
The database can be searched by providing a list of pairs:
sage: l = [m for n in range(1, 4) for m in PerfectMatchings(2*n)] sage: findstat([(m, m.number_of_nestings()) for m in l], depth=0) # optional -- internet 0: St000041 (quality [100, 100])
>>> from sage.all import * >>> l = [m for n in range(Integer(1), Integer(4)) for m in PerfectMatchings(Integer(2)*n)] >>> findstat([(m, m.number_of_nestings()) for m in l], depth=Integer(0)) # optional -- internet 0: St000041 (quality [100, 100])
or a dictionary:
sage: findstat({m: m.number_of_nestings() for m in l}, depth=0) # optional -- internet 0: St000041 (quality [100, 100])
>>> from sage.all import * >>> findstat({m: m.number_of_nestings() for m in l}, depth=Integer(0)) # optional -- internet 0: St000041 (quality [100, 100])
Note however, that the results of these two queries need not compare equal, because we compare queries by the data sent, and the ordering of the data might be different.
Another possibility is to send a collection and a function. In this case, the function is applied to the first few objects of the collection:
sage: findstat("Perfect Matchings", lambda m: m.number_of_nestings(), depth=0) # optional -- internet 0: St000041 (quality [20, 100])
>>> from sage.all import * >>> findstat("Perfect Matchings", lambda m: m.number_of_nestings(), depth=Integer(0)) # optional -- internet 0: St000041 (quality [20, 100])
To search for a distribution, send a list of lists, or a single pair:
sage: PM = PerfectMatchings(10); findstat((PM, [m.number_of_nestings() for m in PM]), depth=0) # optional -- internet 0: St000042 (quality [100, 100]) 1: St000041 (quality [9, 100])
>>> from sage.all import * >>> PM = PerfectMatchings(Integer(10)); findstat((PM, [m.number_of_nestings() for m in PM]), depth=Integer(0)) # optional -- internet 0: St000042 (quality [100, 100]) 1: St000041 (quality [9, 100])
Alternatively, specify the
distribution
parameter:sage: findstat(12, distribution=lambda m: m.number_of_nestings(), depth=0) # optional -- internet 0: St000041 (quality [100, 100]) 1: St000042 (quality [100, 100])
>>> from sage.all import * >>> findstat(Integer(12), distribution=lambda m: m.number_of_nestings(), depth=Integer(0)) # optional -- internet 0: St000041 (quality [100, 100]) 1: St000042 (quality [100, 100])
Note that there is a limit,
FINDSTAT_MAX_VALUES
, on the number of elements that may be submitted to FindStat, which is currently 1000. Therefore, the interface tries to truncate queries appropriately, but this may be impossible, especially with distribution searches:sage: PM = PerfectMatchings(12); PM.cardinality() # optional -- internet 10395 sage: findstat((PM, [1 for m in PM])) # optional -- internet Traceback (most recent call last): ... ValueError: E016: The statistic finder was unable to perform a search on your data. The following errors have occured: You passed too few elements (0 < 3) to FindStat!
>>> from sage.all import * >>> PM = PerfectMatchings(Integer(12)); PM.cardinality() # optional -- internet 10395 >>> findstat((PM, [Integer(1) for m in PM])) # optional -- internet Traceback (most recent call last): ... ValueError: E016: The statistic finder was unable to perform a search on your data. The following errors have occured: <BLANKLINE> You passed too few elements (0 < 3) to FindStat!
Finally, we can also retrieve all statistics with a given domain:
sage: findstat("Cc0024") # optional -- internet Set of combinatorial statistics with domain Cc0024: Binary words in FindStat sage: findstat(domain='Cores') # optional -- internet Set of combinatorial statistics with domain Cc0013: Cores in FindStat
>>> from sage.all import * >>> findstat("Cc0024") # optional -- internet Set of combinatorial statistics with domain Cc0024: Binary words in FindStat >>> findstat(domain='Cores') # optional -- internet Set of combinatorial statistics with domain Cc0013: Cores in FindStat