A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483

Offset: 0

Author

Gus Wiseman, Dec 05 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {2}     {3}        {4}           {5}              {6}
       {11}    {12}       {13}          {14}             {15}
       {1}{1}  {111}      {22}          {23}             {24}
               {1}{1}{1}  {112}         {113}            {33}
                          {1111}        {122}            {114}
                          {2}{2}        {1112}           {123}
                          {11}{11}      {11111}          {222}
                          {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
                                                         {1122}
                                                         {3}{3}
                                                         {11112}
                                                         {111111}
                                                         {12}{12}
                                                         {2}{2}{2}
                                                         {111}{111}
                                                         {11}{11}{11}
                                                         {1}{1}{1}{1}{1}{1}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

For just constant sums we have A305551, ranked by A326534.

For just constant lengths we have A319066, ranked by A320324.

The version for set partitions is A327899.

For distinct instead of constant lengths and sums we have A358832.

The version for twice-partitions is A358833.

A001970 counts multiset partitions of integer partitions.

A063834 counts twice-partitions, strict A296122.

Cf. A000219, A007425, A141199, A327908, A356932, A358831.

Programs

Mathematica

Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]

PARI

P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022

Formula

a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022