A375407 Numbers k such that the k-th composition in standard order (row k of A066099) matches both of the dashed patterns 23-1 and 1-32.

421, 649, 802, 809, 837, 843, 933, 1289, 1299, 1330, 1445, 1577, 1602, 1605, 1617, 1619, 1669, 1673, 1675, 1685, 1686, 1687, 1701, 1826, 1833, 1861, 1867, 1957, 2469, 2569, 2577, 2579, 2597, 2598, 2599, 2610, 2658, 2661, 2674, 2697, 2850, 2857, 2885, 2891

Offset

1,1

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:

(1) the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, and

(2) the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders.

Links

Table of n, a(n) for n=1..44.

Wikipedia, Permutation pattern.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Formula

Intersection of A375138 and A375137.

Example

Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
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 sequence together with corresponding compositions begins:
   421: (1,2,3,2,1)
   649: (2,4,3,1)
   802: (1,3,4,2)
   809: (1,3,2,3,1)
   837: (1,2,4,2,1)
   843: (1,2,3,2,1,1)
   933: (1,1,2,3,2,1)
  1289: (2,5,3,1)
  1299: (2,4,3,1,1)
  1330: (2,3,1,3,2)
  1445: (2,1,2,3,2,1)
  1577: (1,4,2,3,1)
  1602: (1,3,5,2)
  1605: (1,3,4,2,1)
  1617: (1,3,2,4,1)
  1619: (1,3,2,3,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 1000], MatchQ[stc[#], {___, y_, z_, ___, x_, ___}/; x<y<z] && MatchQ[stc[#], {___, x_, ___, z_, y_, ___}/; x<y<z]&]

Crossrefs

The non-dashed version is the intersection of A335482 and A335480.

Compositions of this type are counted by A375297.

For leaders of identical runs we have A375408, counted by A332834.

A003242 counts anti-runs, ranks A333489.

A011782 counts compositions.

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

A335486 ranks compositions matching 21, reverse A335485.

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. A188919, A189076, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375137, A375139, A374768.

Sequence in context: A031784 A268859 A051648 * A217497 A333640 A146358

Adjacent sequences: A375404 A375405 A375406 * A375408 A375409 A375410

Keyword

nonn

Author

Gus Wiseman, Aug 23 2024

Status

approved