A332253 -id:A332253 - cp's OEIS frontend

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

Gus Wiseman, Jul 12 2019

nonn

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

			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}}

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.

Row sums of A332253.

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

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

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}]

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

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

A332260 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n whose union is a k-set where each part has a different size.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 2, 0, 2, 5, 3, 2, 0, 3, 11, 12, 6, 3, 0, 4, 26, 39, 27, 11, 4, 0, 5, 40, 79, 67, 37, 14, 5, 0, 6, 68, 170, 184, 116, 55, 19, 6, 0, 8, 122, 407, 543, 417, 219, 91, 28, 8, 0, 10, 232, 1082, 1911, 1760, 1052, 459, 159, 42, 10

Offset: 0

Andrew Howroyd, Feb 08 2020

T(n,k) is the number of nonequivalent nonnegative integer matrices with total sum n and k nonzero rows with distinct column sums up to permutation of rows and columns.

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 2,   3,   2;
  0, 2,   5,   3,   2;
  0, 3,  11,  12,   6,   3;
  0, 4,  26,  39,  27,  11,   4;
  0, 5,  40,  79,  67,  37,  14,  5;
  0, 6,  68, 170, 184, 116,  55, 19,  6;
  0, 8, 122, 407, 543, 417, 219, 91, 28, 8;
  ...
The T(4,2) = 5 multiset partitions are:
  {{1,1,2,2}}, {{1,2,2,2}}, {{1},{1,2,2}}, {{1},{2,2,2}}, {{1},{1,1,2}}.
These correspond with the following matrices:
   [2]  [1]  [1 1]  [1 0]  [1 2]
   [2]  [3]  [0 2]  [0 3]  [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k=1 is A000009.
Main diagonal is A000009.
Row sums are A326026.
Cf. A332253.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); prod(j=1, #u, 1 + u[j]*x^j + O(x*x^n))/if(!#p, 1, prod(i=1, p[#p], i^v[i]*v[i]!))}
M(n)={my(v=vector(n+1)); for(i=0, n, my(s=0); forpart(p=i, s+=D(p,n)); v[1+i]=Col(s)); Mat(vector(#v, i, v[i]-if(i>1, v[i-1])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

cp's OEIS Frontend

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

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