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A367212
Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.
24
1, 1, 1, 2, 3, 5, 6, 11, 15, 22, 30, 43, 58, 80, 106, 143, 186, 248, 318, 417, 530, 684, 863, 1103, 1379, 1741, 2162, 2707, 3339, 4145, 5081, 6263, 7640, 9357, 11350, 13822, 16692, 20214, 24301, 29300, 35073, 42085, 50208, 59981, 71294, 84866, 100509, 119206
OFFSET
0,4
COMMENTS
Or, partitions whose length is a subset-sum of the parts.
EXAMPLE
The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
The a(1) = 1 through a(8) = 15 partitions:
(1) (11) (21) (22) (32) (42) (52) (62)
(111) (211) (221) (321) (322) (332)
(1111) (311) (2211) (331) (431)
(2111) (3111) (421) (521)
(11111) (21111) (2221) (2222)
(111111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 10}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
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A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971/A364534 count certain types of sum-full subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237668 counts sum-full partitions, sum-free A237667.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, complement A366320.
A365543 counts partitions of n with a subset-sum k, strict A365661.
Sequence in context: A038196 A039849 A039896 * A180336 A034407 A280218
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 11 2023
STATUS
approved