AUTHORS:
Return the set of all Sidon- sets that have elements less than or equal
to
.
A Sidon- set is a set of positive integers
such
that any integer
can be obtain at most
times as sums of unordered pairs of
elements of
(the two elements are not necessary distinct):
INPUT:
OUTPUT:
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
Return the set of all Sidon- sets that have elements less than or equal
to
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}}