cp's OEIS Frontend

A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.

%I A325256 #7 Apr 18 2019 16:54:29
%S A325256 1,1,1,2,3,10,12,12,44,128,228,422,968,1750,420,2100
%N A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size.
%C A325256 A multiset is normal if its union is an initial interval of positive integers.
%C A325256 The adjusted frequency depth of a multiset is 0 if the multiset is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the multiset {1,1,2,2,3} has adjusted frequency depth 5 because we have {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}. The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n).
%e A325256 The a(1) = 1 through a(7) = 12 multisets:
%e A325256   {1}  {12}  {112}  {1123}  {11123}  {111123}  {1112234}
%e A325256              {122}  {1223}  {11223}  {111234}  {1112334}
%e A325256                     {1233}  {11233}  {112345}  {1112344}
%e A325256                             {11234}  {122223}  {1122234}
%e A325256                             {12223}  {122234}  {1123334}
%e A325256                             {12233}  {122345}  {1123444}
%e A325256                             {12234}  {123333}  {1222334}
%e A325256                             {12333}  {123334}  {1222344}
%e A325256                             {12334}  {123345}  {1223334}
%e A325256                             {12344}  {123444}  {1223444}
%e A325256                                      {123445}  {1233344}
%e A325256                                      {123455}  {1233444}
%t A325256 nn=10;
%t A325256 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A325256 fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
%t A325256 mfdm=Table[Max@@fdadj/@allnorm[n],{n,0,nn}];
%t A325256 Table[Length[Select[allnorm[n],fdadj[#]==mfdm[[n+1]]&]],{n,0,nn}]
%Y A325256 Cf. A011784, A181819, A182857, A225486, A323014, A323023, A325238, A325254, A325258, A325277, A325278, A325280, A325282, A325283.
%K A325256 nonn,more
%O A325256 0,4
%A A325256 _Gus Wiseman_, Apr 18 2019