cp's OEIS Frontend

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

%I A326517 #13 Sep 23 2023 15:11:43
%S A326517 1,1,2,12,28,140,956,3520,17792,111600,1144400,4884064,34907936,
%T A326517 214869920,1881044032,25687617152,139175009920,1098825972608,
%U A326517 8770328141888,74286112885504,784394159958848,15114871659653952,92392468773724544,889380453354852416,7652770202041529856
%N A326517 Number of normal multiset partitions of weight n where each part has a different size.
%C A326517 A multiset partition is normal if it covers an initial interval of positive integers.
%H A326517 Andrew Howroyd, <a href="/A326517/b326517.txt">Table of n, a(n) for n = 0..200</a>
%H A326517 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e A326517 The a(0) = 1 through a(3) = 12 normal multiset partitions:
%e A326517   {}  {{1}}  {{1,1}}  {{1,1,1}}
%e A326517              {{1,2}}  {{1,1,2}}
%e A326517                       {{1,2,2}}
%e A326517                       {{1,2,3}}
%e A326517                       {{1},{1,1}}
%e A326517                       {{1},{1,2}}
%e A326517                       {{1},{2,2}}
%e A326517                       {{1},{2,3}}
%e A326517                       {{2},{1,1}}
%e A326517                       {{2},{1,2}}
%e A326517                       {{2},{1,3}}
%e A326517                       {{3},{1,2}}
%p A326517 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A326517       add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
%p A326517     end:
%p A326517 a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
%p A326517 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 23 2023
%t A326517 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326517 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A326517 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A326517 Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]
%o A326517 (PARI)
%o A326517 R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
%o A326517 seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ _Andrew Howroyd_, Feb 07 2020
%Y A326517 Row sums of A332253.
%Y A326517 Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.
%K A326517 nonn
%O A326517 0,3
%A A326517 _Gus Wiseman_, Jul 12 2019
%E A326517 Terms a(8) and beyond from _Andrew Howroyd_, Feb 07 2020