A325278 -id:A325278 - cp's OEIS frontend

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

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

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

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