cp's OEIS Frontend

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

%I A386580 #11 Aug 13 2025 10:21:09
%S A386580 1,1,1,3,3,5,12,13,20,27,64,71,108,145,206,412
%N A386580 Number of normal multisets of size n having a permutation with all distinct run lengths.
%C A386580 A multiset is normal iff it covers an initial interval of positive integers.
%C A386580 Conjecture: Also the number of normal multisets of size n having a disjoint family of strict integer partitions, one of each multiplicity.
%e A386580 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).
%e A386580 The a(n) multisets for n = 1..7:
%e A386580   (1)  (11)  (111)  (1111)  (11111)  (111111)  (1111111)
%e A386580              (112)  (1112)  (11112)  (111112)  (1111112)
%e A386580              (122)  (1222)  (11122)  (111122)  (1111122)
%e A386580                             (11222)  (111222)  (1111222)
%e A386580                             (12222)  (111223)  (1111223)
%e A386580                                      (111233)  (1111233)
%e A386580                                      (112222)  (1112222)
%e A386580                                      (112223)  (1122222)
%e A386580                                      (112333)  (1122223)
%e A386580                                      (122222)  (1123333)
%e A386580                                      (122233)  (1222222)
%e A386580                                      (122333)  (1222233)
%e A386580                                                (1223333)
%t A386580 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A386580 nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
%t A386580 Table[Length[Select[allnorm[n],nodrm[#]!={}&]],{n,0,5}]
%Y A386580 For integer partitions we appear to have A239455, ranks A351294 or A381432.
%Y A386580 For weakly decreasing multiplicities we appear to have A383708.
%Y A386580 The complement is counted by A386581, see A383710 (ranks A382912).
%Y A386580 A000041 counts integer partitions, strict A000009.
%Y A386580 A032020 counts normal multisets with distinct multiplicities, increasing A000009.
%Y A386580 A098859 counts partitions with distinct multiplicities, compositions A242882.
%Y A386580 Cf. A025065, A047966, A048767, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.
%K A386580 nonn,more
%O A386580 0,4
%A A386580 _Gus Wiseman_, Aug 07 2025