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A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

%I A367094 #6 Nov 08 2023 10:45:15
%S A367094 0,1,1,1,2,2,1,5,3,3,8,4,9,1,17,6,16,1,2,24,7,33,4,9,46,11,52,3,18,1,
%T A367094 4,64,12,91,6,38,3,15,1,1,107,17,138,9,68,2,28,2,12,0,2,147,19,219,12,
%U A367094 117,6,56,3,34,2,9,0,3
%N A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.
%F A367094 T(n,1) = A108917(n).
%e A367094 The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
%e A367094 The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
%e A367094 Triangle begins:
%e A367094     0   1
%e A367094     1   1
%e A367094     2   2   1
%e A367094     5   3   3
%e A367094     8   4   9   1
%e A367094    17   6  16   1   2
%e A367094    24   7  33   4   9
%e A367094    46  11  52   3  18   1   4
%e A367094    64  12  91   6  38   3  15   1   1
%e A367094   107  17 138   9  68   2  28   2  12   0   2
%e A367094   147  19 219  12 117   6  56   3  34   2   9   0   3
%e A367094 Row n = 4 counts the following partitions:
%e A367094   (8)     (44)        (431)      (221111)
%e A367094   (71)    (3311)      (422)
%e A367094   (62)    (2222)      (4211)
%e A367094   (611)   (11111111)  (41111)
%e A367094   (53)                (3221)
%e A367094   (521)               (32111)
%e A367094   (5111)              (311111)
%e A367094   (332)               (22211)
%e A367094                       (2111111)
%t A367094 t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
%t A367094 Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]
%Y A367094 Row sums w/o the first column are A002219, ranks A357976, strict A237258.
%Y A367094 Column k = 0 is A006827.
%Y A367094 Row sums are A058696.
%Y A367094 Column k = 1 is A108917.
%Y A367094 The corresponding rank statistic is A357879 (without empty rows).
%Y A367094 A000041 counts integer partitions, strict A000009.
%Y A367094 A182616 counts partitions of 2n that do not contain n, ranks A366321.
%Y A367094 A182616 counts partitions of 2n with at least one odd part, ranks A366530.
%Y A367094 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A367094 A304792 counts subset-sums of partitions, rank statistic A299701.
%Y A367094 A365543 counts partitions of n with a submultiset summing to k.
%Y A367094 Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.
%K A367094 nonn,tabf
%O A367094 0,5
%A A367094 _Gus Wiseman_, Nov 07 2023