

A335479


Numbers k such that the kth composition in standard order (A066099) matches the pattern (1,2,3).


12



52, 104, 105, 108, 116, 180, 200, 208, 209, 210, 211, 212, 216, 217, 220, 232, 233, 236, 244, 308, 328, 360, 361, 364, 372, 400, 401, 404, 408, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 428, 432, 433, 434, 435, 436, 440, 441, 444, 456, 464, 465, 466
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OFFSET

1,1


COMMENTS

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



EXAMPLE

The sequence of terms together with the corresponding compositions begins:
52: (1,2,3)
104: (1,2,4)
105: (1,2,3,1)
108: (1,2,1,3)
116: (1,1,2,3)
180: (2,1,2,3)
200: (1,3,4)
208: (1,2,5)
209: (1,2,4,1)
210: (1,2,3,2)
211: (1,2,3,1,1)
212: (1,2,2,3)
216: (1,2,1,4)
217: (1,2,1,3,1)
220: (1,2,1,1,3)


MATHEMATICA

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


CROSSREFS

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
Nonunimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Other permutations:
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.


KEYWORD

nonn


AUTHOR



STATUS

approved



