A358330 By concatenating the standard compositions of each part of the a(n)-th standard composition, we get a weakly increasing sequence.

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 18, 19, 24, 25, 26, 28, 30, 31, 32, 36, 38, 39, 40, 42, 50, 51, 56, 57, 58, 60, 62, 63, 64, 72, 73, 74, 76, 78, 79, 96, 100, 102, 103, 104, 106, 114, 115, 120, 121, 122, 124, 126, 127, 128, 129, 130, 136, 146, 147

Offset: 1

Author

Gus Wiseman, Nov 10 2022

Keywords

• nonn

Comments

Note we shorten the language, "the k-th composition in standard order," to "the standard composition of k."

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.

Examples

			The terms together with their standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  18: (3,2)
  19: (3,1,1)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
For example, the 532,488-th composition is (6,10,4), with standard compositions ((1,2),(2,2),(3)), with weakly increasing concatenation (1,2,2,2,3), so 532,488 is in the sequence.
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

Standard compositions are listed by A066099.

Indices of rows of A357135 (ranked by A357134) that are weakly increasing.

Cf. A000120, A001511, A029931, A048896, A058891, A070939, A333766, A335404, A357137, A357186.

Programs

Mathematica

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