A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.

1, 1, 0, 2, 3, 6, 20, 65, 134, 482, 1562, 4974, 15466, 51768, 179055, 631737, 2216757, 7905325, 28768472, 106852116, 402255207, 1532029660, 5902839974, 23041880550, 91129833143, 364957188701, 1478719359501, 6058859894440, 25100003070184, 105123020009481, 445036528737301

Offset: 0

Author

Gus Wiseman, Feb 07 2021

Keywords

• nonn

Comments

We define a multiset partition to be twice-balanced if all of the following are equal:

(1) the number of parts;

(2) the number of distinct vertices;

(3) the greatest size of a part.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 6 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{2,3,3}}
            {{2},{1,2}}  {{1,2},{1,2}}  {{1},{2},{2,3,3}}
                         {{1,2},{2,2}}  {{1},{2},{3,3,3}}
                                        {{1},{3},{2,3,3}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}
		

Crossrefs

The co-balanced version is A319616.

The singly balanced version is A340600.

The cross-balanced version is A340651.

The version for factorizations is A340655.

A007716 counts non-isomorphic multiset partitions.

A007718 counts non-isomorphic connected multiset partitions.

A303975 counts distinct prime factors in prime indices.

A316980 counts non-isomorphic strict multiset partitions.

Other balance-related sequences:

- A047993 counts balanced partitions.

- A106529 lists balanced numbers.

- A340596 counts co-balanced factorizations.

- A340653 counts balanced factorizations.

- A340657/A340656 list numbers with/without a twice-balanced factorization.

Cf. A001055, A001221, A001222, A007717, A316983, A320663, A339888, A340597, A340599, A340654.

Programs

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))}
G(m,n,k,y=1)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, y^t*subst(x*Polrev(K(q, t, min(k,n\t))), x, x^t)/t, O(x*x^n)))); s/m!}
seq(n)={Vec(1 + sum(k=1,n, polcoef(G(k,n,k,y) - G(k-1,n,k,y) - G(k,n,k-1,y) + G(k-1,n,k-1,y), k, y)))} \\ Andrew Howroyd, Jan 15 2024

Extensions

a(11) onwards from Andrew Howroyd, Jan 15 2024