A326517 Number of normal multiset partitions of weight n where each part has a different size.

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Author

Gus Wiseman, Jul 12 2019

Keywords

• nonn

Comments

A multiset partition is normal if it covers an initial interval of positive integers.

Examples

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Crossrefs

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

PARI

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Extensions

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020