This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A367212 #15 Nov 12 2023 21:51:51
%S A367212 1,1,1,2,3,5,6,11,15,22,30,43,58,80,106,143,186,248,318,417,530,684,
%T A367212 863,1103,1379,1741,2162,2707,3339,4145,5081,6263,7640,9357,11350,
%U A367212 13822,16692,20214,24301,29300,35073,42085,50208,59981,71294,84866,100509,119206
%N A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.
%C A367212 Or, partitions whose length is a subset-sum of the parts.
%e A367212 The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
%e A367212 The a(1) = 1 through a(8) = 15 partitions:
%e A367212 (1) (11) (21) (22) (32) (42) (52) (62)
%e A367212 (111) (211) (221) (321) (322) (332)
%e A367212 (1111) (311) (2211) (331) (431)
%e A367212 (2111) (3111) (421) (521)
%e A367212 (11111) (21111) (2221) (2222)
%e A367212 (111111) (3211) (3221)
%e A367212 (4111) (3311)
%e A367212 (22111) (4211)
%e A367212 (31111) (22211)
%e A367212 (211111) (32111)
%e A367212 (1111111) (41111)
%e A367212 (221111)
%e A367212 (311111)
%e A367212 (2111111)
%e A367212 (11111111)
%t A367212 Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]
%Y A367212 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.
%Y A367212 sum-full sum-free comb-full comb-free
%Y A367212 -------------------------------------------
%Y A367212 partitions: A367212* A367213 A367218 A367219
%Y A367212 strict: A367214 A367215 A367220 A367221
%Y A367212 subsets: A367216 A367217 A367222 A367223
%Y A367212 ranks: A367224 A367225 A367226 A367227
%Y A367212 A000041 counts partitions, strict A000009.
%Y A367212 A002865 counts partitions whose length is a part, complement A229816.
%Y A367212 A088809/A093971/A364534 count certain types of sum-full subsets.
%Y A367212 A108917 counts knapsack partitions, non-knapsack A366754.
%Y A367212 A126796 counts complete partitions, incomplete A365924.
%Y A367212 A237668 counts sum-full partitions, sum-free A237667.
%Y A367212 A304792 counts subset-sums of partitions, strict A365925.
%Y A367212 Triangles:
%Y A367212 A008284 counts partitions by length, strict A008289.
%Y A367212 A365381 counts sets with a subset summing to k, complement A366320.
%Y A367212 A365543 counts partitions of n with a subset-sum k, strict A365661.
%Y A367212 Cf. A000700, A238628, A363225, A364272, A365658, A365918.
%K A367212 nonn
%O A367212 0,4
%A A367212 _Gus Wiseman_, Nov 11 2023