A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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

Author

Gus Wiseman, Nov 11 2023

Keywords

• nonn

Comments

Or, partitions whose length is a subset-sum of the parts.

Examples

			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)
		

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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partitions: A367212* A367213 A367218 A367219

strict: A367214 A367215 A367220 A367221

subsets: A367216 A367217 A367222 A367223

ranks: A367224 A367225 A367226 A367227

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.

Cf. A000700, A238628, A363225, A364272, A365658, A365918.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]