A011784 -id:A011784 - cp's OEIS frontend

A061894 Lionel-Levine-sequence generated by (2,0).

0, 2, 2, 4, 6, 13, 35, 171, 1934, 97151, 52942129, 1435382350480, 21191828466255176653

Offset: 0

Frank Ellermann, May 13 2001

			a(3) = 4: (2,0),(2,2),(1,1,2,2),(1,1,2,2,3,4)

Cf. A011784, A061892, A061895.

a(10) from Naohiro Nomoto, May 10 2002

a(11) and a(12) from Michael Anthony Keyes, May 12 2021

A327662 Length of shortest word of frequency depth n.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 7, 10, 14, 21, 31, 45, 66, 99

Offset: 1

Ludovic Schwob, Sep 21 2019

The frequency depth of a word is the number of times one must take the multiset of multiplicities to reach the singleton (1), without rearranging it.

For example, the word (11213331) has frequency depth 7: (11213331) -> (21131) -> (1211) -> (112) -> (21) -> (11) -> (2) -> (1).

			The shortest words of frequency depth 10 are of the form (112122112112122122112) and (112122122112112122112), up to substitution and reflection.

Cf. A011784.

Incorrect terms removed by Samuel B. Reid, Aug 25 2021

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 18 2019

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).

			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}

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

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}]

cp's OEIS Frontend

A061894 Lionel-Levine-sequence generated by (2,0).

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A327662 Length of shortest word of frequency depth n.

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

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Mathematica