A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 12, 4, 3, 3, 3, 3, 4, 3, 4, 12, 4, 3, 12, 4, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6, 4, 3, 6, 3, 3, 6, 10, 10, 4, 3, 12, 6, 12, 3, 10, 10, 12, 4, 12, 3, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6

Offset: 0

Author

Gus Wiseman, Jun 20 2020

Keywords

• nonn
• more

Comments

These patterns comprise the basis of the class of patterns generated by this composition.

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).

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.

Examples

			The bases of classes generated by (), (1), (2,1,1), (3,1,2), (2,1,2,1), and (1,2,1), corresponding to n = 0, 1, 11, 38, 45, 13, are the respective columns below.
  (1)  (1,1)  (1,2)    (1,1)    (1,1,1)    (1,1,1)
       (1,2)  (1,1,1)  (1,2,3)  (1,1,2)    (1,1,2)
       (2,1)  (2,2,1)  (1,3,2)  (1,2,2)    (1,2,2)
              (3,2,1)  (2,1,3)  (1,2,3)    (1,2,3)
                       (2,3,1)  (1,3,2)    (1,3,2)
                       (3,2,1)  (2,1,3)    (2,1,1)
                                (2,3,1)    (2,1,2)
                                (3,1,2)    (2,1,3)
                                (3,2,1)    (2,2,1)
                                (2,2,1,1)  (2,3,1)
                                           (3,1,2)
                                           (3,2,1)
		

Links

• 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

Crossrefs

Patterns matched by standard compositions are counted by A335454.

Patterns matched by compositions of n are counted by A335456(n).

The version for Heinz numbers of partitions is A335550.

Patterns are counted by A000670 and ranked by A333217.

Knapsack compositions are counted by A325676 and ranked by A333223.

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

Cf. A056986, A124767, A124770, A124771, A269134, A333218, A333257, A335549.