

A335509


Number of patterns of length n matching the pattern (1,1,2).


10



0, 0, 0, 1, 15, 181, 2163, 27133, 364395, 5272861, 82289163, 1383131773, 24978057195, 483269202781, 9987505786443, 219821796033853, 5137810967933355, 127169580176271901, 3324712113052429323, 91585136315240091133, 2652142325158529483115, 80562824634615270041821
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OFFSET

0,5


COMMENTS

Also the number of (1,2,1)matching patterns of length n.
Also the number of (2,1,2)matching patterns of length 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



FORMULA

E.g.f.: 1/(2exp(x))  (22*x+x^2)/(2*(1x)^2).  Andrew Howroyd, Dec 31 2020


EXAMPLE

The a(3) = 1 through a(4) = 15 patterns:
(1,1,2) (1,1,1,2)
(1,1,2,1)
(1,1,2,2)
(1,1,2,3)
(1,1,3,2)
(1,2,1,2)
(1,2,1,3)
(1,2,2,3)
(1,3,1,2)
(2,1,1,2)
(2,1,1,3)
(2,1,2,3)
(2,2,1,3)
(2,2,3,1)
(3,1,1,2)


MATHEMATICA

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], MatchQ[#, {___, x_, ___, x_, ___, y_, ___}/; x<y]&]], {n, 0, 6}]


PROG

(PARI) seq(n)={Vec(serlaplace(1/(2exp(x + O(x*x^n)))  (22*x+x^2)/(2*(1x)^2)), (n+1))} \\ Andrew Howroyd, Dec 31 2020


CROSSREFS

The complement A001710 is the avoiding version.
Compositions matching this pattern are counted by A335470 and ranked by A335476.
Permutations of prime indices matching this pattern are counted by A335446.
Patterns matching the pattern (1,1) are counted by A019472.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Patterns matching (1,2,3) are counted by A335515.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



