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A325256 Number of normal multisets of size n whose adjusted frequency depth is the maximum for multisets of that size. 0
1, 1, 1, 2, 3, 10, 12, 12, 44, 128, 228, 422, 968, 1750, 420, 2100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

LINKS

Table of n, a(n) for n=0..15.

EXAMPLE

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}

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}]

CROSSREFS

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

Sequence in context: A193652 A092986 A295577 * A302388 A006092 A015859

Adjacent sequences:  A325252 A325253 A325254 * A325258 A325259 A325260

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Apr 18 2019

STATUS

approved

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Last modified August 20 08:27 EDT 2019. Contains 326143 sequences. (Running on oeis4.)