A335454 - OEIS

A335454

Number of normal patterns matched by the n-th composition in standard order (A066099).

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 6, 5, 5, 2, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 10, 9, 9, 3, 6, 5, 9, 4, 9, 10, 12, 5, 9, 7, 13, 7, 12, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 6, 10, 9, 9, 3, 5, 6, 8, 5

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OFFSET

0,2

COMMENTS

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. 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).

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.

LINKS

Table of n, a(n) for n=0..84.

Wikipedia, Permutation pattern

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

EXAMPLE

The a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:

0: 1: 11: 111: 211: 121: 2111: 2311: 1211: 2121:

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() () () () () () () () () ()

(1) (1) (1) (1) (1) (1) (1) (1) (1)

(11) (11) (11) (11) (11) (11) (11) (11)