A071075 Number of permutations that avoid the generalized pattern 132-4.

1, 1, 2, 6, 23, 107, 585, 3671, 25986, 204738, 1776327, 16824237, 172701135, 1909624371, 22626612450, 285982186662, 3840440707485, 54603776221965, 819424594880559, 12942757989763101, 214626518776190178, 3728112755679416898, 67692934780306842501, 1282399636333412178531, 25303124674163685176793

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..460

Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.

Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 1.

Andrew M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.

Andrew M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.

Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.

Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Formula

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = 1/(1 - int(exp(-t^2/2), t=0..y)).

a(n) ~ c * d^n * n! / n^f, where d = 1/A240885 = 1/(sqrt(2)*InverseErf(sqrt(2/Pi))) = 0.7839769312035474991242486548698125357473282..., f = 1.2558142944089303287268746534354522944538722816671534535062816..., c = 0.2242410644782853722452053227678681810005068... . - Vaclav Kotesovec, Aug 23 2014

Let b(n) = A111004(n) = number of permutations of [n] that avoid the consecutive pattern 132. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] - Petros Hadjicostas, Nov 01 2019

Maple

A(y) := 1/(1-int(exp(-t^2/2), t=0..y)); B(x) := exp(int(A(y), y=0..x)); series(B(x), x=0, 30);

Mathematica

CoefficientList[Series[E^(Integrate[1/(1-Integrate[E^(-t^2/2), {t, 0, y}]), {y, 0, x}]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 23 2014 *)

Prog

(PARI)
N=66; x='x+O('x^N);
A=1/(1-intformal(exp(-x^2/2)));
egf=exp(intformal(A));
Vec(serlaplace(egf))
\\ Joerg Arndt, Aug 28 2014

Crossrefs

Cf. A071088, A071076, A071077, A111004.

Sequence in context: A336071 A200403 A113226 * A007555 A101053 A155857

Adjacent sequences: A071072 A071073 A071074 * A071076 A071077 A071078

Keyword

nonn

Author

Sergey Kitaev, May 26 2002

Extensions

Link and a(11)-a(20) from Andrew Baxter, May 17 2011

Typo in first formula corrected by Vaclav Kotesovec, Aug 23 2014

Status

approved