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Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).
56

%I #13 Jun 29 2020 17:10:48

%S 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,

%T 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,

%U 12,4,3,3,3,3,4,3,3,6

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

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

%C 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).

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e 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.

%e (1) (1,1) (1,2) (1,1) (1,1,1) (1,1,1)

%e (1,2) (1,1,1) (1,2,3) (1,1,2) (1,1,2)

%e (2,1) (2,2,1) (1,3,2) (1,2,2) (1,2,2)

%e (3,2,1) (2,1,3) (1,2,3) (1,2,3)

%e (2,3,1) (1,3,2) (1,3,2)

%e (3,2,1) (2,1,3) (2,1,1)

%e (2,3,1) (2,1,2)

%e (3,1,2) (2,1,3)

%e (3,2,1) (2,2,1)

%e (2,2,1,1) (2,3,1)

%e (3,1,2)

%e (3,2,1)

%Y Patterns matched by standard compositions are counted by A335454.

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

%Y The version for Heinz numbers of partitions is A335550.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y Knapsack compositions are counted by A325676 and ranked by A333223.

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

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

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Jun 20 2020