Sidon sets and their generalizations, Sidon \(g\)-sets¶
AUTHORS:
Martin Raum (07-25-2011)
- sage.combinat.sidon_sets.sidon_sets(N, g=1)¶
Return the set of all Sidon-\(g\) sets that have elements less than or equal to \(N\).
A Sidon-\(g\) set is a set of positive integers \(A \subset [1, N]\) such that any integer \(M\) can be obtain at most \(g\) times as sums of unordered pairs of elements of \(A\) (the two elements are not necessary distinct):
\[\#\{ (a_i, a_j) | a_i, a_j \in A, a_i + a_j = M,a_i \leq a_j \} \leq g\]INPUT:
\(N\) – A positive integer.
\(g\) – A positive integer (default: \(1\)).
OUTPUT:
A Sage set with categories whose element are also set of integers.
EXAMPLES:
sage: S = sidon_sets(3, 2) sage: sorted(S, key=str) [{1, 2, 3}, {1, 2}, {1, 3}, {1}, {2, 3}, {2}, {3}, {}] sage: S.cardinality() 8 sage: S.category() Category of finite sets sage: sid = S.an_element() sage: sid {2} sage: sid.category() Category of finite sets
- sage.combinat.sidon_sets.sidon_sets_rec(N, g=1)¶
Return the set of all Sidon-\(g\) sets that have elements less than or equal to \(N\) without checking the arguments. This internal function should not be call directly by user.