login
Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).
3

%I #9 Jan 17 2023 18:22:22

%S 1,0,1,0,4,2,0,10,8,3,0,33,48,18,5,0,91,204,118,32,7,0,298,959,743,

%T 266,58,11,0,910,4193,4334,1927,519,94,15,0,3017,18947,25305,13992,

%U 4407,966,154,22,0,9945,84798,145033,97947,36410,9023,1679,236,30

%N Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

%H Andrew Howroyd, <a href="/A330472/b330472.txt">Table of n, a(n) for n = 0..350</a>

%e Triangle begins:

%e 1

%e 0 1

%e 0 4 2

%e 0 10 8 3

%e 0 33 48 18 5

%e 0 91 204 118 32 7

%e 0 298 959 743 266 58 11

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

%e {{1}{12}} {{1}}{{1}{1}}

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

%e {{2}{11}} {{1}}{{2}{3}}

%e {{1}{1}{1}} {{2}}{{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(A,k,x)*x^k + O(x*x^n), sExp(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 17 2023

%Y Row sums are A318566.

%Y Column k = 1 is A007716 (for n > 0).

%Y Column k = n is A000041.

%Y Partitions of partitions of partitions are A007713.

%Y Twice-factorizations are A050336.

%Y If this is the 3-dimensional version, the 2-dimensional version is A317533.

%Y See A330473 for a variation.

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

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Dec 19 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 17 2023