cp's OEIS Frontend

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

%I A367214 #8 Nov 13 2023 08:46:33
%S A367214 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,
%T A367214 97,113,132,153,178,205,238,272,315,360,413,471,539,613,698,792,899,
%U A367214 1018,1153,1302,1470,1658,1867,2100,2362,2652,2974,3335,3734,4178,4672
%N A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.
%C A367214 These partitions have Heinz numbers A367224 /\ A005117.
%e A367214 The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
%e A367214 The a(1) = 1 through a(10) = 5 strict partitions:
%e A367214   (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
%e A367214                            (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
%e A367214                                              (5,2,1)  (5,3,1)  (6,3,1)
%e A367214                                                       (6,2,1)  (7,2,1)
%e A367214                                                                (4,3,2,1)
%t A367214 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]
%Y A367214 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 A367214                sum-full   sum-free   comb-full  comb-free
%Y A367214               -------------------------------------------
%Y A367214   partitions:  A367212    A367213    A367218    A367219
%Y A367214   strict:      A367214*   A367215    A367220    A367221
%Y A367214   subsets:     A367216    A367217    A367222    A367223
%Y A367214   ranks:       A367224    A367225    A367226    A367227
%Y A367214 A000041 counts integer partitions, strict A000009.
%Y A367214 A088809/A093971/A364534 count certain types of sum-full subsets.
%Y A367214 A188431 counts complete strict partitions, incomplete A365831.
%Y A367214 A240855 counts strict partitions whose length is a part, complement A240861.
%Y A367214 A275972 counts strict knapsack partitions, non-strict A108917.
%Y A367214 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A367214 A365925 counts subset-sums of strict partitions, non-strict A304792.
%Y A367214 Triangles:
%Y A367214 A008289 counts strict partitions by length, non-strict A008284.
%Y A367214 A365661 counts strict partitions with a subset-sum k, non-strict A365543.
%Y A367214 A365832 counts strict partitions by subset-sums, non-strict A365658.
%Y A367214 Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.
%K A367214 nonn
%O A367214 0,7
%A A367214 _Gus Wiseman_, Nov 12 2023