cp's OEIS Frontend

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

%I A340652 #12 Jan 15 2024 18:57:24
%S A340652 1,1,0,2,3,6,20,65,134,482,1562,4974,15466,51768,179055,631737,
%T A340652 2216757,7905325,28768472,106852116,402255207,1532029660,5902839974,
%U A340652 23041880550,91129833143,364957188701,1478719359501,6058859894440,25100003070184,105123020009481,445036528737301
%N A340652 Number of non-isomorphic twice-balanced multiset partitions of weight n.
%C A340652 We define a multiset partition to be twice-balanced if all of the following are equal:
%C A340652 (1) the number of parts;
%C A340652 (2) the number of distinct vertices;
%C A340652 (3) the greatest size of a part.
%e A340652 Non-isomorphic representatives of the a(1) = 1 through a(5) = 6 multiset partitions (empty column indicated by dot):
%e A340652   {{1}}  .  {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{2,3,3}}
%e A340652             {{2},{1,2}}  {{1,2},{1,2}}  {{1},{2},{2,3,3}}
%e A340652                          {{1,2},{2,2}}  {{1},{2},{3,3,3}}
%e A340652                                         {{1},{3},{2,3,3}}
%e A340652                                         {{2},{3},{1,2,3}}
%e A340652                                         {{3},{3},{1,2,3}}
%o A340652 (PARI)
%o A340652 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A340652 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 A340652 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 A340652 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!}
%o A340652 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
%Y A340652 The co-balanced version is A319616.
%Y A340652 The singly balanced version is A340600.
%Y A340652 The cross-balanced version is A340651.
%Y A340652 The version for factorizations is A340655.
%Y A340652 A007716 counts non-isomorphic multiset partitions.
%Y A340652 A007718 counts non-isomorphic connected multiset partitions.
%Y A340652 A303975 counts distinct prime factors in prime indices.
%Y A340652 A316980 counts non-isomorphic strict multiset partitions.
%Y A340652 Other balance-related sequences:
%Y A340652 - A047993 counts balanced partitions.
%Y A340652 - A106529 lists balanced numbers.
%Y A340652 - A340596 counts co-balanced factorizations.
%Y A340652 - A340653 counts balanced factorizations.
%Y A340652 - A340657/A340656 list numbers with/without a twice-balanced factorization.
%Y A340652 Cf. A001055, A001221, A001222, A007717, A316983, A320663, A339888, A340597, A340599, A340654.
%K A340652 nonn
%O A340652 0,4
%A A340652 _Gus Wiseman_, Feb 07 2021
%E A340652 a(11) onwards from _Andrew Howroyd_, Jan 15 2024