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

0, 1, 11, 119, 5615, 251871

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..5.

Example

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)

Mathematica

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}]

Crossrefs

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.

Sequence in context: A125468 A163447 A254600 * A060499 A164828 A060498

Adjacent sequences: A354903 A354904 A354905 * A354907 A354908 A354909

Keyword

nonn,more

Author

Gus Wiseman, Jun 23 2022

Status

approved