A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616

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(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9
		

Crossrefs

Without distinct sums we have A055887.

Twice-partitions of this type are counted by A279786.

For distinct blocks instead of sums we have A304969.

Without constant blocks we have A326519.

Factorizations of this type are counted by A381635.

For strict instead of constant blocks we have A381718.

For equal instead of distinct block-sums we have A382204.

For equal block-sums and strict 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.

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

Normal multiset partitions: A035310, A116540, A255906, A317532.

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

Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

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

Extensions

a(14)-a(26) from Christian Sievers, Apr 04 2025