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

%I A237194 #21 Nov 22 2023 12:25:59
%S A237194 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,
%T A237194 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,
%U A237194 6,7,7,7,4,7,7,7,6,7,15,8,7,8,8,8,8,8
%N A237194 Triangular array:  T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.
%H A237194 Clark Kimberling, <a href="/A237194/b237194.txt">Table of n, a(n) for n = 1..1000</a>
%F A237194 T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.
%e A237194 First 13 rows:
%e A237194 1
%e A237194 0 1
%e A237194 1 1 2
%e A237194 1 0 1 2
%e A237194 1 1 1 1 3
%e A237194 2 2 1 2 2 4
%e A237194 2 2 2 2 2 2 5
%e A237194 3 2 3 1 3 2 3 6
%e A237194 3 3 4 3 3 4 3 3 8
%e A237194 5 4 5 4 3 4 5 4 5 10
%e A237194 5 5 5 5 5 5 5 5 5 5 12
%e A237194 7 6 7 7 7 4 7 7 7 6 7 15
%e A237194 8 7 8 8 8 8 8 8 8 8 7 8 18
%e A237194 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].
%t A237194 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
%t A237194 u = Flatten[%]  (* _Peter J. C. Moses_, Feb 04 2014 *)
%t A237194 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* _Gus Wiseman_, Nov 16 2023 *)
%Y A237194 Column k = n is A000009.
%Y A237194 Column k = 2 is A015744.
%Y A237194 Column k = 1 is A025147.
%Y A237194 The non-strict complement is obtained by adding zeros after A046663.
%Y A237194 Diagonal n = 2k is A237258.
%Y A237194 Row sums are A284640.
%Y A237194 For subsets instead of partitions we have A365381.
%Y A237194 The non-strict version is obtained by removing column k = 0 from A365543.
%Y A237194 Including column k = 0 gives A365661.
%Y A237194 The complement is obtained by adding zeros after A365663.
%Y A237194 Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.
%K A237194 nonn,tabl,easy
%O A237194 1,6
%A A237194 _Clark Kimberling_, Feb 05 2014