A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

1, 1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 8, 15, 9, 14, 15, 17, 13, 22, 14, 25, 21, 23, 19, 34, 24, 29, 28, 37, 27, 45, 29, 44, 38, 43, 43, 59, 40, 51, 48, 69, 48, 71, 52, 73, 69, 72, 61, 93, 72, 91, 77, 99, 78, 105, 95, 119, 95, 113, 96, 146, 107, 126, 123, 151, 130

Offset

0,3

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

Links

Christian Sievers, Table of n, a(n) for n = 0..25000

Formula

G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - Christian Sievers, Apr 05 2025

Example

The a(1) = 1 through a(6) = 7 multiset partitions:
  {1} {11}   {111}     {1111}       {11111}         {111111}
      {1}{1} {2}{11}   {11}{11}     {2}{11}{11}     {111}{111}
             {1}{1}{1} {2}{2}{11}   {2}{2}{2}{11}   {22}{1111}
                       {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
                                                    {2}{2}{11}{11}
                                                    {2}{2}{2}{2}{11}
                                                    {1}{1}{1}{1}{1}{1}
The a(1) = 1 through a(7) = 5 factorizations:
  2  4    8      16       32         64           128
     2*2  3*4    4*4      3*4*4      8*8          3*4*4*4
          2*2*2  3*3*4    3*3*3*4    9*16         3*3*3*4*4
                 2*2*2*2  2*2*2*2*2  4*4*4        3*3*3*3*3*4
                                     3*3*4*4      2*2*2*2*2*2*2
                                     3*3*3*3*4
                                     2*2*2*2*2*2

Mathematica

allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#], SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])], {n, 0, 5}]

Prog

(PARI) h(s, x)=my(t=0, p=1, k=1); while(s%k==0, p*=1/(1-x^(s/k))-1; t+=p; k+=1); t
lista(n)=Vec(1+sum(s=1, n, h(s, x+O(x*x^n)))) \\ Christian Sievers, Apr 05 2025

Crossrefs

Without a common sum we have A055887.

Twice-partitions of this type are counted by A279789.

Without constant blocks we have A326518.

For distinct block-sums and strict blocks we have A381718.

Factorizations of this type are counted by A381995.

For distinct instead of equal block-sums we have A382203.

For strict instead of constant blocks we have A382429.

A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.

A001055 count multiset partitions of prime indices, strict A045778.

A089259 counts set multipartitions of integer partitions.

A255906 counts normal multiset partitions, row sums of A317532.

A321469 counts multiset partitions with distinct block-sums, ranks A326535.

Normal multiset partitions: A035310, A304969, A356945.

Set multipartitions: A116540, A270995, A296119, A318360.

Set multipartitions with distinct sums: A279785, A381806, A381870.

Constant blocks with distinct sums: A381635, A381636, A381716.

Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.

Sequence in context: A347700 A049988 A079247 * A325588 A244903 A342337

Adjacent sequences: A382201 A382202 A382203 * A382205 A382206 A382207

Keyword

nonn

Author

Gus Wiseman, Mar 26 2025

Extensions

Terms a(16) and beyond from Christian Sievers, Apr 04 2025

Status

approved