A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

50, 98, 101, 114, 178, 194, 196, 197, 202, 203, 210, 226, 229, 242, 306, 324, 354, 357, 370, 386, 388, 389, 393, 394, 395, 402, 404, 405, 406, 407, 418, 421, 434, 450, 452, 453, 458, 459, 466, 482, 485, 498, 562, 610, 613, 626, 644, 649, 690, 706, 708, 709

Offset: 1

Author

Gus Wiseman, Aug 09 2024

Keywords

• nonn

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.

These are also numbers k such that the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders, where the leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

The reverse version (A375138) ranks compositions matching the dashed pattern 23-1.

Examples

			Composition 102 is (1,3,1,2), which matches 1-3-2 but not 1-32.
Composition 210 is (1,2,3,2), which matches 1-32 but not 132.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The terms together with corresponding compositions begin:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  210: (1,2,3,2)
  226: (1,1,4,2)
  229: (1,1,3,2,1)
  242: (1,1,1,3,2)
		

Links

• Wikipedia, Permutation pattern.

Crossrefs

The complement is too dense, but counted by A189076.

The non-dashed version is A335480, reverse A335482.

For leaders of identical runs we have A335485, reverse A335486.

For identical leaders we have A374633, counted by A374631.

Compositions of this type are counted by A374636.

For distinct leaders we have A374768, counted by A374632.

The reverse version is A375138, counted by A374636.

For leaders of strictly increasing runs we have A375139, counted by A375135.

Matching 1-21 also gives A375295, counted by A375140 (complement A188920).

A003242 counts anti-runs, ranks A333489.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Leader is A065120.

- Parts are listed by A066099, reverse A228351.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Run-length transform is A333627, sum A070939.

- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.

- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.

Cf. A056823, A106356, A188919, A238343, A333213, A373948, A373953, A374634, A374635, A374637, A375123, A375296.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x