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
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