A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 174, 175, 184, 186, 187, 232, 238, 239, 248, 250, 251, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047

Offset

1,2

Comments

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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).

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=1..57.

Example

The terms together with their binary expansions and corresponding compositions begin:
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   7:      111  (1,1,1)
   8:     1000  (4)
  10:     1010  (2,2)
  11:     1011  (2,1,1)
  14:     1110  (1,1,2)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  31:    11111  (1,1,1,1,1)
  32:   100000  (6)
  36:   100100  (3,3)
  39:   100111  (3,1,1,1)
  42:   101010  (2,2,2)
  46:   101110  (2,1,1,2)
  59:   111011  (1,1,2,1,1)
  60:   111100  (1,1,1,3)
  63:   111111  (1,1,1,1,1,1)

Mathematica

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

Crossrefs

The version for partitions is A353844.

The trajectory length is A353854, firsts A072639, for partitions A353841.

The last part of the trajectory is A353855, for partitions A353842.

These compositions are counted by A353858.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order.

A318928 gives runs-resistance of binary expansion.

A325268 counts partitions by omicron, rank statistic A304465.

A333627 ranks the run-lengths of standard compositions.

A351014 counts distinct runs in standard compositions, firsts A351015.

A353840-A353846 pertain to partition run-sum trajectory.

A353847 represents composition run-sum transformation, partitions A353832.

A353853-A353859 pertain to composition run-sum trajectory.

A353932 lists run-sums of standard compositions.

Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.

Sequence in context: A306677 A351596 A354908 * A353848 A330722 A220969

Adjacent sequences: A353854 A353855 A353856 * A353858 A353859 A353860

Keyword

nonn

Author

Gus Wiseman, Jun 01 2022

Status

approved