A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840

Offset: 0

Author

Gus Wiseman, Dec 14 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(6) = 14 sets:
  {(1)}  {(2)}   {(3)}    {(4)}       {(5)}      {(6)}
         {(11)}  {(21)}   {(22)}      {(32)}     {(33)}
                 {(111)}  {(31)}      {(41)}     {(42)}
                          {(211)}     {(221)}    {(51)}
                          {(1111)}    {(311)}    {(222)}
                          {(2),(11)}  {(2111)}   {(321)}
                                      {(11111)}  {(411)}
                                                 {(2211)}
                                                 {(3111)}
                                                 {(21111)}
                                                 {(111111)}
                                                 {(3),(21)}
                                                 {(3),(111)}
                                                 {(21),(111)}
		

Crossrefs

This is the constant-sum case of A261049, ordered A358906.

The version for all different sums is A271619, ordered A336342.

Allowing repetition gives A305551, ordered A279787.

The version for compositions instead of partitions is A358904.

A001970 counts multisets of partitions.

A034691 counts multisets of compositions, ordered A133494.

A098407 counts sets of compositions, ordered A358907.

Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

Programs

Mathematica

Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]

PARI

a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022

Formula

a(n) = Sum_{d|n} binomial(A000041(d),n/d).