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Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.
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%I #8 Jan 18 2023 12:19:31

%S 1,0,1,0,2,4,0,3,8,10,0,5,28,38,33,0,7,56,146,152,91,0,11,138,474,786,

%T 628,298,0,15,268,1388,3117,3808,2486,910,0,22,570,3843,11830,19147,

%U 18395,9986,3017,0,30,1072,10094,40438,87081,110164,86388,39889,9945

%N Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

%C As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.

%H Andrew Howroyd, <a href="/A330473/b330473.txt">Table of n, a(n) for n = 0..350</a>

%e Triangle begins:

%e 1

%e 0 1

%e 0 2 4

%e 0 3 8 10

%e 0 5 28 38 33

%e 0 7 56 146 152 91

%e 0 11 138 474 786 628 298

%e For example, row n = 3 counts the following multiset partitions:

%e {{111}} {{1}{11}} {{1}{1}{1}}

%e {{112}} {{1}{12}} {{1}{1}{2}}

%e {{123}} {{1}{23}} {{1}{2}{3}}

%e {{2}{11}} {{1}}{{1}{1}}

%e {{1}}{{11}} {{1}}{{1}{2}}

%e {{1}}{{12}} {{1}}{{2}{3}}

%e {{1}}{{23}} {{2}}{{1}{1}}

%e {{2}}{{11}} {{1}}{{1}}{{1}}

%e {{1}}{{1}}{{2}}

%e {{1}}{{2}}{{3}}

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}

%o M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}

%o { my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Jan 18 2023

%Y Row sums are A318566.

%Y Column k = 1 is A000041 (for n > 0).

%Y Column k = n is A007716.

%Y Partitions of partitions of partitions are A007713.

%Y Twice-factorizations are A050336.

%Y The 2-dimensional version is A317533.

%Y See A330472 for a variation.

%Y Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Dec 20 2019

%E Terms a(36) and beyond from _Andrew Howroyd_, Jan 18 2023