A386580 Number of normal multisets of size n having a permutation with all distinct run lengths.

1, 1, 1, 3, 3, 5, 12, 13, 20, 27, 64, 71, 108, 145, 206, 412

Offset: 0

Author

Gus Wiseman, Aug 07 2025

Keywords

• nonn
• more

Comments

A multiset is normal iff it covers an initial interval of positive integers.

Conjecture: Also the number of normal multisets of size n having a disjoint family of strict integer partitions, one of each multiplicity.

Examples

			The normal multiset m = {1,1,1,2,2,2} has permutation (1,2,2,2,1,1) with run lengths (1,3,2), so m is counted under a(6).
The a(n) multisets for n = 1..7:
  (1)  (11)  (111)  (1111)  (11111)  (111111)  (1111111)
             (112)  (1112)  (11112)  (111112)  (1111112)
             (122)  (1222)  (11122)  (111122)  (1111122)
                            (11222)  (111222)  (1111222)
                            (12222)  (111223)  (1111223)
                                     (111233)  (1111233)
                                     (112222)  (1112222)
                                     (112223)  (1122222)
                                     (112333)  (1122223)
                                     (122222)  (1123333)
                                     (122233)  (1222222)
                                     (122333)  (1222233)
                                               (1223333)
		

Crossrefs

For integer partitions we appear to have A239455, ranks A351294 or A381432.

For weakly decreasing multiplicities we appear to have A383708.

The complement is counted by A386581, see A383710 (ranks A382912).

A000041 counts integer partitions, strict A000009.

A032020 counts normal multisets with distinct multiplicities, increasing A000009.

A098859 counts partitions with distinct multiplicities, compositions A242882.

Cf. A025065, A047966, A048767, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.

Programs

Mathematica

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
Table[Length[Select[allnorm[n],nodrm[#]!={}&]],{n,0,5}]