A335456 Number of normal patterns matched by compositions of n.

1, 2, 5, 12, 32, 84, 211, 556, 1446, 3750, 9824, 25837, 67681, 178160, 468941, 1233837, 3248788

Offset

0,2

Comments

A composition of n is a finite sequence of positive integers summing to n.

We define a 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..16.

Wikipedia, Permutation pattern

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

Example

The 8 compositions of 4 together with the a(4) = 32 patterns they match:
  4:   31:   13:   22:   211:   121:   112:   1111:
-----------------------------------------------------
  ()   ()    ()    ()    ()     ()     ()     ()
  (1)  (1)   (1)   (1)   (1)    (1)    (1)    (1)
       (21)  (12)  (11)  (11)   (11)   (11)   (11)
                         (21)   (12)   (12)   (111)
                         (211)  (21)   (112)  (1111)
                                (121)

Mathematica

mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]], {y, Join@@Permutations/@IntegerPartitions[n]}], {n, 0, 8}]

Crossrefs

References found in the link are not all included here.

The version for standard compositions is A335454.

The contiguous case is A335457.

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 patterns avoided by a standard composition are counted by A335465.

Cf. A034691, A056986, A106356, A108917, A124770, A269134, A329744, A333224, A335458.

Sequence in context: A162434 A277863 A039809 * A345200 A173611 A292211

Adjacent sequences: A335453 A335454 A335455 * A335457 A335458 A335459

Keyword

nonn,more

Author

Gus Wiseman, Jun 16 2020

Extensions

a(14)-a(16) from Jinyuan Wang, Jun 26 2020

Status

approved