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A365660
Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.
3
1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037
OFFSET
0,4
COMMENTS
Are n = 1, 2, 4 the only n such that none of these partitions has 1?
Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?
EXAMPLE
The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2) (2,2) (4,2) (4,2,2) (4,3,3) (6,4,2) (6,5,3)
(5,1) (2,2,2,2) (4,4,2) (6,5,1) (8,4,2)
(2,2,2) (6,2,2) (4,4,2,2) (8,5,1)
(8,1,1) (6,2,2,2) (9,3,2)
(4,2,2,2) (4,2,2,2,2) (9,4,1)
(2,2,2,2,2) (2,2,2,2,2,2) (10,3,1)
(11,2,1)
(4,4,4,2)
(5,3,3,3)
(6,4,2,2)
(8,2,2,2)
(11,1,1,1)
(4,4,2,2,2)
(6,2,2,2,2)
(4,2,2,2,2,2)
(2,2,2,2,2,2,2)
MATHEMATICA
msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]], {n, 0, 10}]
PROG
(Python)
from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
c = 0
for p in partitions(n<<1):
q, s = list(Counter(p).elements()), set()
for l in range(1, len(q)+1):
for k in multiset_combinations(q, l):
s.add(sum(k))
if len(s) > n:
break
else:
continue
break
if len(s)==n:
c += 1
return c # Chai Wah Wu, Sep 20 2023
CROSSREFS
For n instead of 2n we have A126796.
Central column n = 2k of A365658.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002219 counts partitions of 2n with a submultiset summing to n.
A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
A122768 counts distinct nonempty submultisets of partitions.
A299701 counts sums of submultisets of prime indices, of partitions A304792.
A364272 counts sum-full strict partitions, sum-free A364349.
A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
Sequence in context: A014686 A053090 A264400 * A309512 A225367 A283479
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 16 2023
EXTENSIONS
a(21)-a(38) from Chai Wah Wu, Sep 20 2023
a(39)-a(43) from Chai Wah Wu, Sep 21 2023
STATUS
approved