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

1, 1, 1, 2, 3, 10, 12, 12, 44, 128, 228, 422, 968, 1750, 420, 2100

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn
• more

Comments

A multiset is normal if its union is an initial interval of positive integers.

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).

Examples

			The a(1) = 1 through a(7) = 12 multisets:
  {1}  {12}  {112}  {1123}  {11123}  {111123}  {1112234}
             {122}  {1223}  {11223}  {111234}  {1112334}
                    {1233}  {11233}  {112345}  {1112344}
                            {11234}  {122223}  {1122234}
                            {12223}  {122234}  {1123334}
                            {12233}  {122345}  {1123444}
                            {12234}  {123333}  {1222334}
                            {12333}  {123334}  {1222344}
                            {12334}  {123345}  {1223334}
                            {12344}  {123444}  {1223444}
                                     {123445}  {1233344}
                                     {123455}  {1233444}
		

Crossrefs

Cf. A011784, A181819, A182857, A225486, A323014, A323023, A325238, A325254, A325258, A325277, A325278, A325280, A325282, A325283.

Programs

Mathematica

nn=10;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
mfdm=Table[Max@@fdadj/@allnorm[n],{n,0,nn}];
Table[Length[Select[allnorm[n],fdadj[#]==mfdm[[n+1]]&]],{n,0,nn}]