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A335469
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Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,2).
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6
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
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OFFSET
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1,3
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COMMENTS
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A composition of n is a finite sequence of positive integers summing to n. 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.
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).
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LINKS
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EXAMPLE
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See A335468 for an example of the complement.
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MATHEMATICA
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stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]];
Select[Range[0, 100], !MatchQ[stc[#], {___, x_, ___, y_, ___, x_, ___}/; x>y]&]
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CROSSREFS
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The complement A335468 is the matching version.
The (1,2,1)-avoiding version is A335467.
These compositions are counted by A335473.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A001710, A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335450, A335456, A335458.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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