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A111053 Number of permutations which avoid the patterns 1324 and (2143 with Bruhat restriction {2<->3}). Also the number of permutations whose graphs are acyclic. 0
1, 2, 6, 22, 89, 379, 1661, 7405, 33367, 151398, 690147, 3156112, 14465746, 66409493, 305232025, 1404129530, 6463476538, 29767212095, 137142651679, 632021380433, 2913316615372, 13431328632593, 61931182541194, 285592218851606, 1317104663887309, 6074682489939359, 28018852961838675, 129239701278757210 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399, Table A.7.

LINKS

Table of n, a(n) for n=1..28.

H. Abe and S. Billey, Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, 2014.

M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.

S. Butler, On permutations which are 1324 and {overline 2143} avoiding, 2005.

S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Haruhisa Enomoto, Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras, arXiv:2002.09205 [math.RT], 2020.

FORMULA

G.f.: ((1-X)*(1-4*X-2*X*X)-(1-5*X)*sqrt(1-4*X))/2/(1-5*X+2*X^2-X^3. - Ralf Stephan, May 09 2007

G.f.: 2 * x * (1 - 4*x - x^2) / ((1 - x) * (1 - 4*x - 2*x^2) + (1 - 5*x) * sqrt(1 - 4*x)). - Michael Somos, Jan 12 2012

G.f. is the power series composition of g.f. A204200 and g.f. A000108 (Catalan) with offset 1. - Michael Somos, Jan 12 2012

Conjecture: n*(n+5)*a(n) +3*(20-13*n-3*n^2)*a(n-1) +2*(11*n^2+40*n-150)*a(n-2) +3*(40-11*n-3*n^2)*a(n-3) +2*(n+6)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Aug 14 2012

EXAMPLE

x + 2*x^2 + 6*x^3 + 22*x^4 + 89*x^5 + 379*x^6 + 1661*x^7 + 7405*x^8 + ...

MATHEMATICA

a = DifferenceRoot[Function[{a, n}, {(4n^2 + 46n + 60)a[n] + (-9n^2 - 105n - 156)a[n+1] + (22n^2 + 256n + 372)a[n+2] + (-9n^2 - 111n - 240)a[n+3] + (n+4)(n+9)a[n+4] == 0, a[1] == 1, a[2] == 2, a[3] == 6, a[4] == 22}]];

Array[a, 28] (* Jean-François Alcover, Dec 17 2018 *)

PROG

(PARI) x='x+O('x^66);

gf=((1-x)*(1-4*x-2*x^2)-(1-5*x)*sqrt(1-4*x))/(2*(1-5*x+2*x^2-x^3));

Vec(gf) /* Joerg Arndt, Jun 25 2011 */

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 * x * (1 - 4*x - x^2) / ((1 - x) * (1 - 4*x - 2*x^2) + (1 - 5*x) * sqrt(1 - 4*x + x * O(x^n))), n))} /* Michael Somos, Jan 12 2012 */

CROSSREFS

Cf. A204200.

Sequence in context: A150267 A271388 A165540 * A165541 A165542 A165543

Adjacent sequences:  A111050 A111051 A111052 * A111054 A111055 A111056

KEYWORD

nonn

AUTHOR

Steve Butler, Oct 06 2005

EXTENSIONS

More terms from Joerg Arndt, Jun 25 2011

STATUS

approved

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Last modified June 23 01:56 EDT 2021. Contains 345394 sequences. (Running on oeis4.)