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
Keywords
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.
- Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, and Haijun Li, Consecutive and quasi-consecutive patterns: des-Wilf classifications and generating functions, arXiv:2502.10128 [math.CO], 2025. See p. 6.
Programs
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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);
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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 *) -
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
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
Extensions
Link and a(11)-a(20) from Andrew Baxter, May 17 2011
Typo in first formula corrected by Vaclav Kotesovec, Aug 23 2014