A316980 Number of non-isomorphic strict multiset partitions of weight n.

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

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

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

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

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

Sequence in context: A248870 A103819 A147484 * A331321 A017929 A017930

Adjacent sequences: A316977 A316978 A316979 * A316981 A316982 A316983

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