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A335466
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Numbers k such that the k-th composition in standard order (A066099) matches (1,2,1).
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4
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13, 25, 27, 29, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 193, 195
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OFFSET
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1,1
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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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The sequence of terms together with the corresponding compositions begins:
13: (1,2,1)
25: (1,3,1)
27: (1,2,1,1)
29: (1,1,2,1)
45: (2,1,2,1)
49: (1,4,1)
51: (1,3,1,1)
53: (1,2,2,1)
54: (1,2,1,2)
55: (1,2,1,1,1)
57: (1,1,3,1)
59: (1,1,2,1,1)
61: (1,1,1,2,1)
77: (3,1,2,1)
82: (2,3,2)
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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 A335467 is the avoiding version.
The (2,1,2)-matching version is A335468.
These compositions are counted by A335470.
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. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335509.
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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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