A335458 - OEIS

A335458

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

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 5, 5, 5, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 7, 8, 9, 3, 5, 5, 8, 4, 8, 7, 11, 5, 8, 7, 11, 7, 11, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 5, 7, 8, 9, 3, 5, 5, 8, 5, 7

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OFFSET

0,2

COMMENTS

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 (normal) 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

Table of n, a(n) for n=0..85.

Wikipedia, Permutation pattern

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

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

FORMULA

a(n) = A335474(n) + 1.

EXAMPLE

The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

MATHEMATICA

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]];

mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];