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

0,3

Links

Table of n, a(n) for n=0..45.

Formula

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

Example

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)}

Mathematica

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

Prog

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

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.

Sequence in context: A066880 A075427 A075426 * A191615 A018606 A117087

Adjacent sequences: A359038 A359039 A359040 * A359042 A359043 A359044

Keyword

nonn

Author

Gus Wiseman, Dec 14 2022

Status

approved