A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 4, 4, 4, 1, 2, 2, 4, 2, 4, 4, 6, 2, 4, 4, 7, 4, 7, 6, 5, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 3, 6, 4, 6, 7, 8, 2, 4, 4, 7, 3, 7, 6, 10, 4, 7, 6, 10, 6, 10, 8, 6, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 4, 6, 4, 6, 7, 8, 2, 4, 4, 7, 4, 6

Offset

0,4

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) = A335458(n) - 1.

Example

The a(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:
      (1)  (1)   (1)    (1)    (1)     (1)     (1)
           (12)  (21)   (12)   (12)    (11)    (12)
                 (321)  (21)   (21)    (12)    (21)
                        (231)  (121)   (21)    (121)
                               (213)   (122)   (123)
                               (2131)  (221)   (212)
                                       (2331)  (1212)
                                               (2123)
                                               (12123)

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]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n], {___, s__, ___}:>{s}]]], {n, 0, 100}]

Crossrefs

The version for Heinz numbers of partitions is A335516(n) - 1.

The non-contiguous version is A335454(n) - 1.

The version allowing empty patterns is A335458.

Patterns are counted by A000670 and ranked by A333217.

The n-th composition has A124771(n) distinct consecutive subsequences.

Knapsack compositions are counted by A325676 and ranked by A333223.

The n-th composition has A334299(n) distinct subsequences.

Minimal avoided patterns are counted by A335465.

Patterns matched by prime indices are counted by A335549.

Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.

Sequence in context: A178677 A366888 A243924 * A333939 A272759 A272760

Adjacent sequences: A335471 A335472 A335473 * A335475 A335476 A335477

Keyword

nonn

Author

Gus Wiseman, Jun 21 2020

Status

approved