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, 8554709

Offset: 0

Author

Gus Wiseman, Jun 16 2020

Keywords

• nonn
• more

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).

Examples

			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)
		

Links

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

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.

Programs

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}]

Extensions

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

a(17) from John Tyler Rascoe, Mar 14 2025