A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

1, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 5, 3, 2, 3, 1, 3, 2, 3, 6, 3, 3, 4, 3, 3, 4, 3, 3, 8, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 7, 6, 7, 7, 7, 4, 7, 7, 7, 6, 7, 15, 8, 7, 8, 8, 8, 8, 8

Offset: 1

Author

Clark Kimberling, Feb 05 2014

Keywords

• nonn
• tabl
• easy

Examples

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Column k = n is A000009.

Column k = 2 is A015744.

Column k = 1 is A025147.

The non-strict complement is obtained by adding zeros after A046663.

Diagonal n = 2k is A237258.

Row sums are A284640.

For subsets instead of partitions we have A365381.

The non-strict version is obtained by removing column k = 0 from A365543.

Including column k = 0 gives A365661.

The complement is obtained by adding zeros after A365663.

Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Programs

Mathematica

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

Formula

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.