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%I #54 Mar 24 2024 03:12:10
%S 1,1,2,6,23,107,585,3671,25986,204738,1776327,16824237,172701135,
%T 1909624371,22626612450,285982186662,3840440707485,54603776221965,
%U 819424594880559,12942757989763101,214626518776190178,3728112755679416898,67692934780306842501,1282399636333412178531,25303124674163685176793
%N Number of permutations that avoid the generalized pattern 132-4.
%H Vaclav Kotesovec, <a href="/A071075/b071075.txt">Table of n, a(n) for n = 0..460</a>
%H Juan S. Auli, <a href="https://search.proquest.com/openview/3f0cef1fbdb016d61e16412e4b855969/1">Pattern Avoidance in Inversion Sequences</a>, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
%H Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/1904.02694">Consecutive Patterns in Inversion Sequences</a>, arXiv:1904.02694 [math.CO], 2019. See Table 1.
%H Andrew M. Baxter, <a href="https://pdfs.semanticscholar.org/2c5d/79e361d3aecb25c380402144177ad7cd9dc8.pdf">Algorithms for Permutation Statistics</a>, Ph. D. Dissertation, Rutgers University, May 2011.
%H Andrew M. Baxter, <a href="https://doi.org/doi:10.7282/T3HQ3Z89">Algorithms for Permutation Statistics</a>, Ph. D. Dissertation, Rutgers University, May 2011.
%H Andrew M. Baxter and Lara K. Pudwell, <a href="http://arxiv.org/abs/1108.2642">Enumeration schemes for dashed patterns</a>, arXiv:1108.2642 [math.CO], 2011.
%H Sergey Kitaev, <a href="https://web.archive.org/web/20040902121307/http://www.ms.uky.edu/~kitaev/index_files/Papers/partial_order_patterns_perm.pdf">Partially Ordered Generalized Patterns</a>, preprint.
%H Sergey Kitaev, <a href="http://dx.doi.org/10.1016/j.disc.2004.03.017">Partially Ordered Generalized Patterns</a>, Discrete Math. 298 (2005), no. 1-3, 212-229.
%F E.g.f.: exp(int(A(y), y=0..x)), where A(y) = 1/(1 - int(exp(-t^2/2), t=0..y)).
%F 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
%F 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
%p 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);
%t 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 *)
%o (PARI)
%o N=66; x='x+O('x^N);
%o A=1/(1-intformal(exp(-x^2/2)));
%o egf=exp(intformal(A));
%o Vec(serlaplace(egf))
%o \\ _Joerg Arndt_, Aug 28 2014
%Y Cf. A071088, A071076, A071077, A111004.
%K nonn
%O 0,3
%A _Sergey Kitaev_, May 26 2002
%E Link and a(11)-a(20) from _Andrew Baxter_, May 17 2011
%E Typo in first formula corrected by _Vaclav Kotesovec_, Aug 23 2014
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