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A321661
Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.
4
1, 1, 1, 4, 4, 7, 22, 25, 40, 58, 186, 204, 347, 478, 734, 2033, 2402, 3814, 5464, 8142, 11058, 30142, 34437, 55940, 77794, 116954, 156465, 229462, 533612, 640544, 994922, 1397896, 2048316, 2778750, 3987432, 5292293, 11921070, 14076550, 21802928, 29917842, 44080285
OFFSET
0,4
COMMENTS
The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, whose nonzero entries are all distinct.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
LINKS
FORMULA
a(n) = Sum_{k>=1} A059849(k)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(6) = 22 multiset partitions:
{{1}} {{11}} {{111}} {{1111}} {{11111}} {{111111}}
{{122}} {{1222}} {{11222}} {{112222}}
{{1}{11}} {{1}{111}} {{12222}} {{122222}}
{{1}{22}} {{1}{222}} {{1}{1111}} {{122333}}
{{11}{111}} {{1}{11111}}
{{11}{222}} {{11}{1111}}
{{1}{2222}} {{1}{11222}}
{{11}{1222}}
{{11}{2222}}
{{112}{222}}
{{11}{2333}}
{{1}{22222}}
{{122}{222}}
{{1}{22333}}
{{122}{333}}
{{2}{11222}}
{{22}{1222}}
{{1}{11}{111}}
{{1}{11}{222}}
{{1}{22}{222}}
{{1}{22}{333}}
{{2}{11}{222}}
PROG
(PARI) \\ here b(n) is A059849(n).
b(n)={sum(k=0, n, stirling(n, k, 1)*sum(i=0, k, stirling(k, i, 2))^2)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 15 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018
STATUS
approved