

A335468


Numbers k such that the kth composition in standard order (A066099) matches the pattern (2,1,2).


3



22, 45, 46, 54, 76, 86, 90, 91, 93, 94, 109, 110, 118, 148, 150, 153, 156, 166, 173, 174, 178, 180, 181, 182, 183, 186, 187, 189, 190, 204, 214, 218, 219, 221, 222, 237, 238, 246, 278, 280, 297, 300, 301, 302, 306, 307, 308, 310, 313, 316, 326, 332, 333, 334
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OFFSET

1,1


COMMENTS

A composition of n is a finite sequence of positive integers summing to n. The kth composition in standard order (graded reverselexicographic, 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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).


LINKS

Table of n, a(n) for n=1..54.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions


EXAMPLE

The sequence together with the corresponding compositions begins:
22: (2,1,2)
45: (2,1,2,1)
46: (2,1,1,2)
54: (1,2,1,2)
76: (3,1,3)
86: (2,2,1,2)
90: (2,1,2,2)
91: (2,1,2,1,1)
93: (2,1,1,2,1)
94: (2,1,1,1,2)
109: (1,2,1,2,1)
110: (1,2,1,1,2)
118: (1,1,2,1,2)
148: (3,2,3)
150: (3,2,1,2)


MATHEMATICA

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


CROSSREFS

The complement A335469 is the avoiding version.
The (1,2,1)matching version is A335466.
These compositions are counted by A335472.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Nonunimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335453, A335456, A335458, A335509.
Sequence in context: A138869 A138872 A003858 * A041956 A041954 A041952
Adjacent sequences: A335465 A335466 A335467 * A335469 A335470 A335471


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jun 16 2020


STATUS

approved



