A322786 -id:A322786 - cp's OEIS frontend

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120

Offset: 0

Gus Wiseman, Dec 26 2018

nonn

A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020

			The a(1) = 1 through a(4) = 29 multiset partitions:
  {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
          {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
         {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
                     {{2},{1,3}}     {{1,1},{1,1}}
                     {{3},{1,2}}     {{1},{1,2,2}}
                    {{1},{1},{1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}    {{1,2},{1,2}}
                                     {{1},{2,3,4}}
                                     {{1,2},{3,4}}
                                     {{1,3},{2,4}}
                                     {{1,4},{2,3}}
                                     {{2},{1,1,2}}
                                     {{2},{1,3,4}}
                                     {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                    {{1},{1},{1,1}}
                                    {{1},{1},{2,2}}
                                    {{1},{2},{1,2}}
                                    {{1},{2},{3,4}}
                                    {{1},{3},{2,4}}
                                    {{1},{4},{2,3}}
                                    {{2},{2},{1,1}}
                                    {{2},{3},{1,4}}
                                    {{2},{4},{1,3}}
                                    {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{3},{4}}

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

Row sums of A322786.

Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]

a(n) = Sum_{d|n} A001055(A002110(n/d)^d).

a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020

a(14)-a(15) from Alois P. Heinz, Jan 16 2019

Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020

A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6

Offset: 1

Gus Wiseman, Dec 26 2018

A multiset partition is uniform if all parts have the same size.

			Triangle begins:
     1
     2    2
     2    2
     5    4    3
     2    2
    27   11    6    4
     2    2
   142   29    8    4
   282   12    3
  1073  101    8    4
The multiset partitions counted under row 6:
  {123456}          {112233}          {111222}          {111111}
  {123}{456}        {112}{233}        {111}{222}        {111}{111}
  {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
  {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
  {126}{345}        {123}{123}        {12}{12}{12}
  {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
  {135}{246}        {11}{23}{23}
  {136}{245}        {12}{12}{33}
  {145}{236}        {12}{13}{23}
  {146}{235}        {13}{13}{22}
  {156}{234}        {1}{1}{2}{2}{3}{3}
  {12}{34}{56}
  {12}{35}{46}
  {12}{36}{45}
  {13}{24}{56}
  {13}{25}{46}
  {13}{26}{45}
  {14}{23}{56}
  {14}{25}{36}
  {14}{26}{35}
  {15}{23}{46}
  {15}{24}{36}
  {15}{26}{34}
  {16}{23}{45}
  {16}{24}{35}
  {16}{25}{34}
  {1}{2}{3}{4}{5}{6}

Andrew Howroyd, Table of n, a(n) for n = 1..482 (rows 1..100)

Row sums are A322785. First column is A038041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.

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]]]];
Table[Length[Select[mps[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]

T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).

More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
Edited by Peter Munn, Mar 05 2025

cp's OEIS Frontend

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

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