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A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

%I A367106 #8 Nov 12 2023 21:51:26
%S A367106 1,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,2,1,1,0,0,0,1,2,1,1,0,0,0,1,3,2,
%T A367106 1,1,0,0,0,0,3,3,2,1,1,0,0,0,0,4,5,3,2,1,1,0,0,0,0,3,5,5,3,2,1,1,0,0,
%U A367106 0,0,4,8,7,5,3,2,1,1,0,0,0,0,2,9,9,7,5
%N A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.
%C A367106 An integer partition of n is complete (ranks A325781) if every integer from 0 to n is the sum of some submultiset of the parts.
%e A367106 Triangle begins:
%e A367106   1
%e A367106   0  1
%e A367106   0  0  1
%e A367106   0  0  1  1
%e A367106   0  0  0  1  1
%e A367106   0  0  0  2  1  1
%e A367106   0  0  0  1  2  1  1
%e A367106   0  0  0  1  3  2  1  1
%e A367106   0  0  0  0  3  3  2  1  1
%e A367106   0  0  0  0  4  5  3  2  1  1
%e A367106   0  0  0  0  3  5  5  3  2  1  1
%e A367106   0  0  0  0  4  8  7  5  3  2  1  1
%e A367106   0  0  0  0  2  9  9  7  5  3  2  1  1
%e A367106   0  0  0  0  2 11 12 11  7  5  3  2  1  1
%e A367106   0  0  0  0  1 11 16 13 11  7  5  3  2  1  1
%e A367106   0  0  0  0  1 14 21 19 15 11  7  5  3  2  1  1
%e A367106 Row n = 11 counts the following partitions (empty columns not shown):
%e A367106   6311  62111  611111  5111111  41111111  311111111  2111111111  11111111111
%e A367106   6221  53111  521111  4211111  32111111  221111111
%e A367106   5321  52211  431111  3311111  22211111
%e A367106   4421  44111  422111  3221111
%e A367106         43211  332111  2222111
%e A367106         42221  322211
%e A367106         33311  222221
%e A367106         33221
%t A367106 nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t A367106 Table[Length[Select[IntegerPartitions[n,{k}],nmz[#]=={}&]],{n,0,15},{k,0,n}]
%Y A367106 Column k appears to have A000325(k) nonzero terms.
%Y A367106 Column sums are A003513.
%Y A367106 Central column T(2n,n) is A007042.
%Y A367106 Row sums are A126796, ranks A325781.
%Y A367106 The strict case is too sparse, row sums A188431 (complement A365831).
%Y A367106 Grouping by maximum instead of length gives A261036.
%Y A367106 A000041 counts integer partitions.
%Y A367106 A108917 counts knapsack partitions, ranks A299702.
%Y A367106 A299701 counts subset-sums of prime indices, firsts A259941.
%Y A367106 A365924 counts incomplete partitions, ranks A365830.
%Y A367106 Cf. A002033, A018818, A046663, A047967, A055932, A276024, A304792, A365921.
%K A367106 nonn,tabl
%O A367106 0,19
%A A367106 _Gus Wiseman_, Nov 09 2023