A386581 Number of normal multisets of size n with no permutation having all distinct run lengths.

0, 0, 1, 1, 5, 11, 20, 51, 108, 229, 448, 953, 1940, 3951, 7986, 15972

Offset: 0

Author

Gus Wiseman, Aug 12 2025

Keywords

• nonn
• more

Comments

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

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 not counted under a(6).
The a(1) = 0 through a(6) = 20 multisets:
  .  (12)  (123)  (1122)  (11123)  (111123)
                  (1123)  (11223)  (111234)
                  (1223)  (11233)  (112233)
                  (1233)  (11234)  (112234)
                  (1234)  (12223)  (112334)
                          (12233)  (112344)
                          (12234)  (112345)
                          (12333)  (122223)
                          (12334)  (122234)
                          (12344)  (122334)
                          (12345)  (122344)
                                   (122345)
                                   (123333)
                                   (123334)
                                   (123344)
                                   (123345)
                                   (123444)
                                   (123445)
                                   (123455)
                                   (123456)
		

Crossrefs

The complement for partitions appears to be A239455, ranks A351294 or A381432.

For integer partitions we appear to have A351293, ranks A351295 or A381433.

For weakly decreasing multiplicities we appear to have A383710, ranks A382912.

The complement is counted by A386580, see A383708.

A032020 counts normal multisets with distinct multiplicities.

A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).

A098859 counts partitions with distinct multiplicities, compositions A242882.

Cf. A000009, A025065, A047966, 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,0,7}]