A022558 Number of permutations of length n avoiding the pattern 1342.

1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, 6411521056, 43478151737, 297864793993, 2059159989914, 14350039389022, 100726680316559, 711630547589023, 5057282786190872, 36132861123763276, 259423620328055093

Offset: 0

Author

Miklos Bona

Keywords

• nonn
• easy

Comments

Differs from A117156 which counts permutations avoiding the *consecutive* pattern 1342. - Ray Chandler, Dec 06 2011

Also, number of permutation of length n avoiding the pattern 3142 (see Stankova (1994) below). - Alexander Burstein, Aug 09 2013

Conjecture: a(n) is the number of permutations of length n simultaneously avoiding patterns 2143 and 415263. - Alexander Burstein, Mar 21 2019

Examples

			a(4) = 23 because obviously all permutations of length 4 with the exception of 1342 avoid 1342.
		

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 768, Th. 12.1.14.
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.48.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
• Miklos Bona, Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps, arXiv:math/9702223 [math.CO], 1997.
• Miklos Bona, Exact enumeration of 1342-avoiding permutations; A close link with labeled trees and planar maps, J. Combinatorial Theory, A80 (1997), 257-272.
• Alexander Burstein and Jay Pantone, Two examples of unbalanced Wilf-equivalence, J. Combin. 6 (2015), no. 1-2, 55-67.
• Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
• A. R. Conway and A. J. Guttmann, On 1324-avoiding permutations, Adv. Appl. Math. 64 (2015), 50-69.
• A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
• Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
• C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint arXiv:1410.2657 [math.CO], 2014.
• Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
• W. Mlotkowski, K. A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 (2015), eq. (36).
• Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), no. 1-3, 291-316.
• Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 11.

Crossrefs

Essentially the same as A004040.

Cf. A117158.

A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

Programs

Maple

a := proc (n) options operator, arrow: (1/2)*(-1)^(n-1)*(7*n^2-3*n-2)+3*(sum((-1)^(n-i)*2^(i+1)*factorial(2*i-4)*binomial(n-i+2, 2)/(factorial(i)*factorial(i-2)), i = 2 .. n)) end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Oct 15 2014

Mathematica

Table[SeriesCoefficient[32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[1/2*(-1)^(n-1) * (-2-3*n+7*n^2) + 1/4*(-1)^n * (1+n) * (-2-13*n+(n+2) * Hypergeometric2F1[-3/2,-n,-2-n,-8]),{n,0,20}] (* Vaclav Kotesovec, Aug 24 2014 *)

PARI

x='x+O('x^66); Vec( 32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)) ) \\ Joerg Arndt, May 04 2013

Formula

a(n) = (7*n^2-3*n-2)/2 * (-1)^(n-1) + 3*Sum_{i=2..n} 2^(i+1) * (2*i-4)!/(i!*(i-2)!) * binomial(n-i+2, 2) * (-1)^(n-i).

G.f.: 32*x/(1 + 20*x - 8*x^2 - (1 - 8*x)^(3/2)). - Emeric Deutsch, Mar 13 2004

Recurrence: n*a(n) = (7*n-22)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 07 2012

a(n) ~ 2^(3*n+6)/(243*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Oct 07 2012

Extensions

Minor edits by Vaclav Kotesovec, Aug 24 2014