OFFSET
0,3
COMMENTS
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).
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
FORMULA
Euler transform of A319557. - Gus Wiseman, Sep 23 2018
EXAMPLE
Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
(111) <-> (111)
(122) <-> (1)(11)
(1)(11) <-> (122)
(1)(22) <-> (1)(22)
(2)(12) <-> (2)(12)
(1)(1)(1) <-> (123)
(1)(2)(2) <-> (1)(23)
(1)(2)(3) <-> (1)(2)(3)
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
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))}
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 18 2018
EXTENSIONS
a(7)-a(10) from Gus Wiseman, Sep 23 2018
Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023
STATUS
approved