A317757 Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.

1, 0, 1, 4, 17, 56, 205, 690, 2446, 8506, 30429, 109449, 402486, 1501424, 5714194, 22132604, 87383864, 351373406, 1439320606, 6003166059, 25488902820, 110125079184, 483987225922, 2162799298162, 9823464989574, 45332196378784, 212459227340403, 1010898241558627, 4881398739414159

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

Cf. A007716, A035310, A255906, A281116, A317073, A317533.

Cf. A317748, A317751, A317752, A317755.

Cf. A319077, A319748, A319755, A319778, A319781, A319790.

Sequence in context: A255526 A264218 A121327 * A183924 A255271 A293798

Adjacent sequences: A317754 A317755 A317756 * A317758 A317759 A317760

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