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

1,6

Links

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

Formula

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

Example

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].

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 *)

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.

Sequence in context: A178948 A203827 A194289 * A143519 A344606 A341592

Adjacent sequences: A237191 A237192 A237193 * A237195 A237196 A237197

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 05 2014

Status

approved