A354912 Numbers k such that the k-th composition in standard order is the sequence of run-sums of some other composition.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 52, 54, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88, 89, 90, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Links

Table of n, a(n) for n=0..65.

Example

The terms and their corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
For example, the 21st composition in standard order (2,2,1) equals the run-sums of (1,1,2,1), so 21 is in the sequence. On the other hand, no composition has run-sums equal to the 29th composition (1,1,2,1), so 29 is not in the sequence.

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], MemberQ[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[Total[stc[#]]], stc[#]]&]

Crossrefs

The standard compositions used here are A066099, run-sums A353847/A353932.

These are the positions of nonzero terms in A354578.

The complement is A354904, counted by A354909.

These compositions are counted by A354910.

A003242 counts anti-run compositions, ranked by A333489.

A124767 counts runs in standard compositions.

A238279 and A333755 count compositions by number of runs.

A351014 counts distinct runs of standard compositions, firsts A351015.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353853-A353859 pertain to composition run-sum trajectory.

Cf. A000120, A029837, A124771, A239312, A333381, A334299, A351597, A353832, A353849, A353860, A354905.

Sequence in context: A143028 A277018 A277008 * A091529 A184967 A379051

Adjacent sequences: A354909 A354910 A354911 * A354913 A354914 A354915

Keyword

nonn

Author

Gus Wiseman, Jun 22 2022

Status

approved