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A325835
Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.
3
0, 0, 1, 2, 3, 5, 9, 10, 14, 22, 30, 33, 46, 52, 74, 107, 101, 123, 171, 182, 225
OFFSET
0,4
COMMENTS
The number of submultisets of a partition is the product of its multiplicities, each plus one. A subset-sum of an integer partition is the sum of some submultiset of its parts. These are partitions with one subset-sum which is the sum of two distinct submultisets, while all others are the sum of only one submultiset.
The Heinz numbers of these partitions are given by A325802.
EXAMPLE
The a(2) = 1 through a(8) = 14 partitions:
(211) (321) (422) (532) (633) (743) (844)
(3111) (431) (541) (642) (752) (853)
(41111) (5221) (651) (761) (862)
(5311) (4332) (7322) (871)
(511111) (5331) (7331) (5443)
(6222) (7421) (7441)
(6411) (7511) (7531)
(33222) (72221) (8332)
(6111111) (74111) (8521)
(71111111) (8611)
(82222)
(83311)
(85111)
(811111111)
For example, the partition (7,5,3,1) has submultisets (), (1), (3), (5), (7), (3,1), (5,1), (5,3), (7,1), (7,3), (7,5), (5,3,1), (7,3,1), (7,5,1), (7,5,3), (7,5,3,1), all of which have different sums except for (5,3) and (7,1), which both sum to 8, so (7,5,3,1) is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==1+Length[Union[Total/@Subsets[#]]]&]], {n, 0, 20, 2}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 29 2019
STATUS
approved