A321661 - OEIS

%I #15 Nov 17 2018 20:56:58

%S 1,1,1,4,4,7,22,25,40,58,186,204,347,478,734,2033,2402,3814,5464,8142,

%T 11058,30142,34437,55940,77794,116954,156465,229462,533612,640544,

%U 994922,1397896,2048316,2778750,3987432,5292293,11921070,14076550,21802928,29917842,44080285

%N Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

%C The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.

%C 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.

%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%H Andrew Howroyd, <a href="/A321661/b321661.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k>=1} A059849(k)*A008289(n,k) for n > 0. - _Andrew Howroyd_, Nov 17 2018

%e Non-isomorphic representatives of the a(1) = 1 through a(6) = 22 multiset partitions:

%e   {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}    {{111111}}

%e                  {{122}}    {{1222}}    {{11222}}    {{112222}}

%e                  {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}

%e                  {{1}{22}}  {{1}{222}}  {{1}{1111}}  {{122333}}

%e                                         {{11}{111}}  {{1}{11111}}

%e                                         {{11}{222}}  {{11}{1111}}

%e                                         {{1}{2222}}  {{1}{11222}}

%e                                                      {{11}{1222}}

%e                                                      {{11}{2222}}

%e                                                      {{112}{222}}

%e                                                      {{11}{2333}}

%e                                                      {{1}{22222}}

%e                                                      {{122}{222}}

%e                                                      {{1}{22333}}

%e                                                      {{122}{333}}

%e                                                      {{2}{11222}}

%e                                                      {{22}{1222}}

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

%e                                                      {{1}{11}{222}}

%e                                                      {{1}{22}{222}}

%e                                                      {{1}{22}{333}}

%e                                                      {{2}{11}{222}}

%o (PARI) \\ here b(n) is A059849(n).

%o b(n)={sum(k=0, n, stirling(n,k,1)*sum(i=0, k, stirling(k,i,2))^2)}

%o 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

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 15 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Nov 16 2018