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

Offset: 0

Author

Gus Wiseman, Jun 14 2020

Keywords

• nonn
• look

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.

Examples

			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:
---------------------------------------------------------------------
  ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
      (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
           (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
                 (111)  (21)   (12)   (21)    (12)    (12)    (12)
                        (211)  (21)   (111)   (21)    (21)    (21)
                               (121)  (211)   (211)   (111)   (121)
                                      (2111)  (231)   (121)   (211)
                                              (2311)  (211)   (212)
                                                      (1211)  (221)
                                                              (2121)
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..8192
• 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.

Crossrefs

References found in the links are not all included here.

Summing over indices with binary length n gives A335456(n).

The contiguous case is A335458.

The version for Heinz numbers of partitions is A335549.

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 A333257(n) distinct subsequence-sums.

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

Minimal avoided patterns are counted by A335465.

Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

Programs

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/@Subsets[stc[n]]]],{n,0,30}]

Python

from itertools import combinations
def comp(n):
    # see A357625
    return
def A335465(n):
    A,B,C = set(),set(),comp(n)
    c = range(len(C))
    for j in c:
        for k in combinations(c, j):
            A.add(tuple(C[i] for i in k))
    for i in A:
        D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
        B.add(tuple(D[v] for v in i))
    return len(B)+1 # John Tyler Rascoe, Mar 12 2025