A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset

0,3

Comments

First differs from A353432 (the consecutive case) in having 0 and 53.

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..50.

Example

The initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y], Length/@Split[y]];
Select[Range[0, 100], rosQ[stc[#]]&]

Crossrefs

The version for partitions is A325755, counted by A325702.

These compositions are counted by A353390.

The recursive version is A353431, counted by A353391.

The consecutive case is A353432, counted by A353392.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order, reverse A228351.

A333769 lists run-lengths of compositions in standard order.

Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.

Statistics of standard compositions:

- Length is A000120, sum A070939.

- Runs are counted by A124767, distinct A351014.

- Subsequences are counted by A334299, consecutive A124770/A124771.

- Runs-resistance is A333628.

Classes of standard compositions:

- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.

- Runs are A272919.

- Golomb rulers are A333222, counted by A169942.

- Knapsack compositions are A333223, counted by A325676.

- Anti-runs are A333489, counted by A003242.

Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

Sequence in context: A181450 A364079 A367576 * A353432 A331997 A087597

Adjacent sequences: A353399 A353400 A353401 * A353403 A353404 A353405

Keyword

nonn

Author

Gus Wiseman, May 15 2022

Status

approved