A116699 Number of permutations of length n which avoid the patterns 123 and 4312.

1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290, 2956, 3757, 4710, 5833, 7145, 8666, 10417, 12420, 14698, 17275, 20176, 23427, 27055, 31088, 35555, 40486, 45912, 51865, 58378, 65485, 73221, 81622, 90725, 100568, 111190, 122631, 134932

Offset

1,2

Comments

Also number of permutations of length n which avoid the patterns 321, 2134 (reverse symmetry); or 321, 1243 (complement symmetry); etc.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.

Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: x*(2*x^3 - 5*x^2 + 3*x - 1)/(x-1)^5.

a(n) = (n^4 + 2*n^3 - 13*n^2 + 34*n)/24. - Franklin T. Adams-Watters, Sep 16 2006

Partial sums of A105163. - Levi R. Self (levi.r.self(AT)gmail.com), Aug 04 2007

Binomial transform of [1, 1, 2, 3, 1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 23 2007

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 5, 13, 30}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
CoefficientList[Series[(2 x^3 - 5 x^2 + 3 x - 1)/(x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 01 2014 *)

Prog

(PARI) for(n=1, 100, print1((n^4 + 2*n^3 - 13*n^2 + 34*n)/24", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 22 2008
(Magma) [(n^4 + 2*n^3 - 13*n^2 + 34*n)/24: n in [1..45]]; // Vincenzo Librandi, Nov 01 2014

Crossrefs

Sequence in context: A122025 A236414 A057873 * A290198 A282153 A054127

Adjacent sequences: A116696 A116697 A116698 * A116700 A116701 A116702

Keyword

nonn,easy

Author

Lara Pudwell, Feb 26 2006

Extensions

Edited by N. J. A. Sloane, Mar 16 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 22 2008

Status

approved