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 A022558 Number of permutations of length n avoiding the pattern 1342. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 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. 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 EXAMPLE a(4) = 23 because obviously all permutations of length 4 with the exception of 1342 avoid 1342. 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 *) PROG (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 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). Sequence in context: A238639 A226995 A301897 * A004040 A216040 A005802 Adjacent sequences: A022555 A022556 A022557 * A022559 A022560 A022561 KEYWORD nonn,easy AUTHOR Miklos Bona EXTENSIONS Minor edits by Vaclav Kotesovec, Aug 24 2014 STATUS approved

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