A325256 - OEIS

A325256

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

%I #7 Apr 18 2019 16:54:29

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

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

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

%C 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 The a(1) = 1 through a(7) = 12 multisets:

%e {1} {12} {112} {1123} {11123} {111123} {1112234}

%e {122} {1223} {11223} {111234} {1112334}

%e {1233} {11233} {112345} {1112344}

%e {11234} {122223} {1122234}

%e {12223} {122234} {1123334}

%e {12233} {122345} {1123444}

%e {12234} {123333} {1222334}

%e {12333} {123334} {1222344}

%e {12334} {123345} {1223334}

%e {12344} {123444} {1223444}

%e {123445} {1233344}

%e {123455} {1233444}

%t nn=10;

%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];

%t mfdm=Table[Max@@fdadj/@allnorm[n],{n,0,nn}];

%t Table[Length[Select[allnorm[n],fdadj[#]==mfdm[[n+1]]&]],{n,0,nn}]

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

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Apr 18 2019