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: S
{{2}, {3}, {1, 2}, {}, {2, 3}, {1}, {1, 3}, {1, 2, 3}}
sage: S.cardinality()
8
sage: S.category()
Category of sets
sage: sid = S.an_element()
sage: sid
{2}
sage: sid.category()
Category of sets

TESTS:

sage: S = sidon_sets(10)
sage: TestSuite(S).run()
sage: Set([1,2,4,8,13]) in sidon_sets(13)
True

The following piece of code computes the first values of the Sloane sequence entitled ‘Length of shortest (or optimal) Golomb ruler with n marks’ with a very dumb algorithm. (sequence identifier A003022):

sage: n = 1
sage: L = []
sage: for i in range(1,19):
...       nb = max([S.cardinality() for S in sidon_sets(i)])
...       if nb > n:
...           L.append(i-1)
...           n = nb
sage: L
[1, 3, 6, 11, 17]

The following tests check that some generalized Sidon sets satisfy the right conditions, using a dumb but exhaustive algorithm:

sage: from itertools import groupby
sage: all(all(l <= 3 for l in map(lambda s: len(list(s[1])), groupby(sorted(a + ap for a in sid for ap in sid if a >= ap), lambda s: s))) for sid in sidon_sets(10, 3))
True
sage: all(all(l <= 5 for l in map(lambda s: len(list(s[1])), groupby(sorted(a + ap for a in sid for ap in sid if a >= ap), lambda s: s))) for sid in sidon_sets(10, 5))
True

Checking of arguments:

sage: sidon_sets(1,1)
{{}, {1}}
sage: sidon_sets(-1,3)
Traceback (most recent call last):
...
ValueError: N must be a positive integer
sage: sidon_sets(1, -3)
Traceback (most recent call last):
...
ValueError: g must be a positive integer
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.

TESTS:

sage: from sage.combinat.sidon_sets import sidon_sets_rec
sage: sidon_sets_rec(3,2)
{{2}, {3}, {1, 2}, {}, {2, 3}, {1}, {1, 3}, {1, 2, 3}}

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