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A335468
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Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,2).
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3
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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
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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 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)
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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 A335469 is the avoiding version.
The (1,2,1)-matching version is A335466.
These compositions are counted by A335472.
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, A335453, 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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