A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Author

Gus Wiseman, Nov 12 2023

Keywords

• nonn

Comments

These partitions have Heinz numbers A367224 /\ A005117.

Examples

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (7,2,1)
                                                               (4,3,2,1)
		

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 integer partitions, strict A000009.

A088809/A093971/A364534 count certain types of sum-full subsets.

A188431 counts complete strict partitions, incomplete A365831.

A240855 counts strict partitions whose length is a part, complement A240861.

A275972 counts strict knapsack partitions, non-strict A108917.

A364272 counts sum-full strict partitions, sum-free A364349.

A365925 counts subset-sums of strict partitions, non-strict A304792.

Triangles:

A008289 counts strict partitions by length, non-strict A008284.

A365661 counts strict partitions with a subset-sum k, non-strict A365543.

A365832 counts strict partitions by subset-sums, non-strict A365658.

Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]