A335518 Number of matching pairs of patterns, the first of length n and the second of length k.

1, 1, 1, 3, 3, 3, 13, 13, 25, 13, 75, 75, 185, 213, 75, 541, 541, 1471, 2719, 2053, 541, 4683, 4683, 13265, 32973, 40367, 22313, 4683, 47293, 47293, 136711, 408265, 713277, 625295, 271609, 47293

Offset

0,4

Comments

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..35.

Wikipedia, Permutation pattern

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

Example

Triangle begins:
     1
     1     1
     3     3     3
    13    13    25    13
    75    75   185   213    75
   541   541  1471  2719  2053   541
  4683  4683 13265 32973 40367 22313  4683
Row n =2 counts the following pairs:
  ()<=(1,1)  (1)<=(1,1)  (1,1)<=(1,1)
  ()<=(1,2)  (1)<=(1,2)  (1,2)<=(1,2)
  ()<=(2,1)  (1)<=(2,1)  (2,1)<=(2,1)

Mathematica

mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[Union[mstype/@Subsets[y, {k}]]], {y, Join@@Permutations/@allnorm[n]}], {n, 0, 5}, {k, 0, n}]

Crossrefs

Columns k = 0 and k = 1 are both A000670.

Row sums are A335517.

Patterns are ranked by A333217.

Patterns matched by a standard composition are counted by A335454.

Patterns contiguously matched by compositions are counted by A335457.

Minimal patterns avoided by a standard composition are counted by A335465.

Patterns matched by prime indices are counted by A335549.

Cf. A011782, A034691, A056986, A124771, A269134, A329744, A333257, A334299.

Sequence in context: A378153 A147823 A341211 * A269347 A183554 A229847

Adjacent sequences: A335515 A335516 A335517 * A335519 A335520 A335521

Keyword

nonn,tabl,more

Author

Gus Wiseman, Jun 23 2020

Status

approved