cp's OEIS Frontend

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

%I A386581 #8 Aug 13 2025 10:20:25
%S A386581 0,0,1,1,5,11,20,51,108,229,448,953,1940,3951,7986,15972
%N A386581 Number of normal multisets of size n with no permutation having all distinct run lengths.
%C A386581 A multiset is normal iff it covers an initial interval of positive integers.
%e A386581 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).
%e A386581 The a(1) = 0 through a(6) = 20 multisets:
%e A386581   .  (12)  (123)  (1122)  (11123)  (111123)
%e A386581                   (1123)  (11223)  (111234)
%e A386581                   (1223)  (11233)  (112233)
%e A386581                   (1233)  (11234)  (112234)
%e A386581                   (1234)  (12223)  (112334)
%e A386581                           (12233)  (112344)
%e A386581                           (12234)  (112345)
%e A386581                           (12333)  (122223)
%e A386581                           (12334)  (122234)
%e A386581                           (12344)  (122334)
%e A386581                           (12345)  (122344)
%e A386581                                    (122345)
%e A386581                                    (123333)
%e A386581                                    (123334)
%e A386581                                    (123344)
%e A386581                                    (123345)
%e A386581                                    (123444)
%e A386581                                    (123445)
%e A386581                                    (123455)
%e A386581                                    (123456)
%t A386581 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A386581 nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
%t A386581 Table[Length[Select[allnorm[n],nodrm[#]=={}&]],{n,0,7}]
%Y A386581 The complement for partitions appears to be A239455, ranks A351294 or A381432.
%Y A386581 For integer partitions we appear to have A351293, ranks A351295 or A381433.
%Y A386581 For weakly decreasing multiplicities we appear to have A383710, ranks A382912.
%Y A386581 The complement is counted by A386580, see A383708.
%Y A386581 A032020 counts normal multisets with distinct multiplicities.
%Y A386581 A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
%Y A386581 A098859 counts partitions with distinct multiplicities, compositions A242882.
%Y A386581 Cf. A000009, A025065, A047966, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.
%K A386581 nonn,more
%O A386581 0,5
%A A386581 _Gus Wiseman_, Aug 12 2025