A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

Offset

0,3

Comments

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.

Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018

From Gus Wiseman, Oct 11 2018: (Start)

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:

{{1,1,1,1}}

{{1,1,2,2}}

{{1,2,3,4}}

{{1},{1,1,1}}

{{1},{1,2,2}}

{{1},{2,3,4}}

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{3,4}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{2},{1,2}}

{{1},{2},{3,4}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{3},{4}}

(End)

Links

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

Formula

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

Example

Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{2},{2}}
  {{1},{2},{3},{3}}
  {{1},{2},{3},{4}}

Mathematica

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

Prog

(PARI) \\ See A318951 for RowSumMats.
a(n)={sumdiv(n, d, RowSumMats(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018

Crossrefs

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

Sequence in context: A365111 A190968 A127416 * A226631 A226634 A105271

Adjacent sequences: A306014 A306015 A306016 * A306018 A306019 A306020

Keyword

nonn

Author

Gus Wiseman, Jun 17 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

Status

approved