A382429 Number of normal multiset partitions of weight n into sets with a common sum.

1, 1, 2, 3, 5, 7, 13, 26, 57, 113, 283, 854, 2401, 6998, 24072, 85061, 308956, 1190518, 4770078, 19949106, 87059592

Offset: 0

Author

Gus Wiseman, Mar 26 2025

Keywords

• nonn
• more

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.

Examples

			The a(1) = 1 through a(6) = 13 partitions:
  {1} {12}   {123}     {1234}       {12345}         {123456}
      {1}{1} {3}{12}   {12}{12}     {24}{123}       {123}{123}
             {1}{1}{1} {14}{23}     {34}{124}       {125}{134}
                       {3}{3}{12}   {3}{12}{12}     {135}{234}
                       {1}{1}{1}{1} {5}{14}{23}     {145}{235}
                                    {3}{3}{3}{12}   {12}{12}{12}
                                    {1}{1}{1}{1}{1} {14}{14}{23}
                                                    {14}{23}{23}
                                                    {16}{25}{34}
                                                    {3}{3}{12}{12}
                                                    {5}{5}{14}{23}
                                                    {3}{3}{3}{3}{12}
                                                    {1}{1}{1}{1}{1}{1}
The corresponding factorizations:
  2  6    30     210      2310       30030
     2*2  5*6    6*6      21*30      30*30
          2*2*2  14*15    35*42      6*6*6
                 5*5*6    5*6*6      66*70
                 2*2*2*2  5*5*5*6    110*105
                          11*14*15   154*165
                          2*2*2*2*2  5*5*6*6
                                     14*14*15
                                     14*15*15
                                     26*33*35
                                     5*5*5*5*6
                                     11*11*14*15
                                     2*2*2*2*2*2
		

Crossrefs

Without the common sum we have A116540 (normal set multipartitions).

Twice-partitions of this type are counted by A279788.

For common sizes instead of sums we have A317583.

Without strict blocks we have A326518, non-strict blocks A326517.

For a common length instead of sum we have A331638.

For distinct instead of equal block-sums we have A381718.

Factorizations of this type are counted by A382080.

For distinct block-sums and constant blocks we have A382203.

For constant instead of strict blocks we have A382204.

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

A001055 count multiset partitions of prime indices, strict A045778.

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

Normal multiset partitions: A035310, A255906, A304969, A317532.

Set multipartitions: A089259, A116539, A270995, A296119, A318360.

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

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

Cf. A000110, A034691, A038041, A050320, A255903, A326520, A381633, A381996, A382214, A382216.

Programs

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@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

Extensions

a(11) from Robert Price, Mar 30 2025

a(12)-a(20) from Christian Sievers, Apr 06 2025