OFFSET
0,4
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
EXAMPLE
Non-isomorphic representatives of the a(4) = 17 multiset partitions:
{1}{234},{2}{111},{2}{113},{11}{22},{11}{23},{12}{34},
{1}{1}{22},{1}{1}{23},{1}{2}{11},{1}{2}{12},{1}{2}{13},{1}{2}{34},{2}{3}{11},
{1}{1}{1}{2},{1}{1}{2}{2},{1}{1}{2}{3},{1}{2}{3}{4}.
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]]; sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Join@@Table[Select[mps[m], Intersection@@#=={}&], {m, strnorm[n]}]]], {n, 6}]
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, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q, n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f, k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t], O(x*x^n) ))/if(k, 1-x^k, 1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 06 2018
EXTENSIONS
a(8)-a(10) from Gus Wiseman, Sep 27 2018
a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, May 30 2023
STATUS
approved