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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 (list; graph; refs; listen; history; text; internal format)
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

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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

CROSSREFS

Cf. A000219, A007716, A008289, A059201, A059849, A114736, A117433, A120733, A321653, A321659, A321660, A321662.

Sequence in context: A336718 A173324 A318243 * A183492 A019159 A019250

Adjacent sequences:  A321658 A321659 A321660 * A321662 A321663 A321664

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 15 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

STATUS

approved

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Last modified September 25 19:58 EDT 2020. Contains 337344 sequences. (Running on oeis4.)