%I #15 Jan 21 2023 17:58:10

%S 1,1,3,8,23,63,197,588,1892,6140,20734,71472,254090,923900,3446572,

%T 13149295,51316445,204556612,832467052,3455533022,14621598811,

%U 63023667027,276559371189,1234802595648,5606647482646,25875459311317,121324797470067,577692044073205

%N Number of non-isomorphic strict multiset partitions of weight n.

%C Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

%C Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

%H Andrew Howroyd, <a href="/A316980/b316980.txt">Table of n, a(n) for n = 0..50</a>

%F Euler transform of A319557. - _Gus Wiseman_, Sep 23 2018

%e Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):

%e (111) <-> (111)

%e (122) <-> (1)(11)

%e (1)(11) <-> (122)

%e (1)(22) <-> (1)(22)

%e (2)(12) <-> (2)(12)

%e (1)(1)(1) <-> (123)

%e (1)(2)(2) <-> (1)(23)

%e (1)(2)(3) <-> (1)(2)(3)

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ _Andrew Howroyd_, Jan 21 2023

%Y Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.

%Y Cf. A316974, A316978, A316979, A316981, A316983.

%Y Cf. A049311, A059201, A319558, A319560, A319567.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 18 2018

%E a(7)-a(10) from _Gus Wiseman_, Sep 23 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 19 2023