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%I #15 Sep 14 2019 12:19:49
%S 1,1,1,3,3,5,13,15,23,33,49,59,83,101,133,281,321,477,655,941,1249,
%T 1795,2241,3039,3867,5047,6257,8063,11459,13891,18165,23149,29975,
%U 37885,49197,61829,89877,109165,145673,185671,246131,310325,408799,514485,668017,871383
%N Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.
%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, with all different entries.
%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="/A321662/b321662.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - _Andrew Howroyd_, Nov 17 2018
%e Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:
%e {{111}} {{1111}} {{11111}} {{111111}} {{1111111}}
%e {{122}} {{1222}} {{11222}} {{112222}} {{1112222}}
%e {{1}{11}} {{1}{111}} {{12222}} {{122222}} {{1122222}}
%e {{1}{1111}} {{122333}} {{1222222}}
%e {{11}{111}} {{1}{11111}} {{1223333}}
%e {{11}{1111}} {{1}{111111}}
%e {{1}{11222}} {{11}{11111}}
%e {{11}{1222}} {{111}{1111}}
%e {{112}{222}} {{1}{112222}}
%e {{122}{222}} {{11}{12222}}
%e {{2}{11222}} {{112}{2222}}
%e {{22}{1222}} {{122}{2222}}
%e {{1}{11}{111}} {{2}{112222}}
%e {{22}{12222}}
%e {{1}{11}{1111}}
%t (* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];
%t (* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];
%t a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];
%t a /@ Range[0, 45] (* _Jean-François Alcover_, Sep 14 2019 *)
%o (PARI) \\ here b(n) is A121860(n).
%o b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}
%o seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); 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, A114736, A117433, A120733, A121860, A321653, A321659, A321660, A321661.
%K nonn
%O 0,4
%A _Gus Wiseman_, Nov 15 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Nov 16 2018
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