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.

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

Author

Gus Wiseman, Dec 26 2018

Keywords

• nonn
• look
• tabf

Comments

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

Examples

			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}
		

Links

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

Crossrefs

Row sums are A322785. First column is A038041.

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

Programs

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

Formula

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

Extensions

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