A125188 Number of Dumont permutations of the first kind of length 2n avoiding the patterns 2413 and 4132. Also number of Dumont permutations of the first kind of length 2n avoiding the patterns 1423 and 3142.

1, 1, 3, 12, 54, 259, 1294, 6655, 34986, 187149, 1015407, 5574829, 30915904, 172933249, 974605751, 5528804444, 31546576802, 180931023589, 1042503934315, 6031773336043, 35030156585236, 204135876541762, 1193291688154639

Offset: 0

Author

Emeric Deutsch, Dec 19 2006

Keywords

• nonn

Links

• A. Burstein, Restricted Dumont permutations, arXiv:math/0402378 [math.CO], 2004
• A. Burstein, Restricted Dumont permutations, Annals of Combinatorics, 9, 2005, 269-280 (Theorem 3.13).
• Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
• Matteo Cervetti and Luca Ferrari, Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset, Annals of Combinatorics (2022).
• Y. Sun and Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832, G_{uud}

Crossrefs

Cf. A125187.

Programs

Maple

C:=(1-sqrt(1-4*x))/2/x: G:=(1+x*C-sqrt(1-x*C-5*x))/2/x/(1+C): Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=0..26);

Mathematica

CoefficientList[Series[(-3+Sqrt[2]*Sqrt[1+Sqrt[1-4*x]-10*x] + Sqrt[1-4*x])/(2*(-1+Sqrt[1-4*x]-2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

Formula

G.f.=[1+xC(x)-sqrt(1-xC(x)-5x)]/[2x(1+C(x))], where C(x)=(1-sqrt(1-4x))/(2x) is the Catalan function.

D-finite with recurrence 32*(n-1)*(2*n-1)*(n+1)*a(n) +8*(-148*n^3+461*n^2-367*n+14)*a(n-1) +4*(2197*n^3-13436*n^2+25653*n-14694)*a(n-2) +2*(-16868*n^3+159415*n^2-483427*n+468080)*a(n-3) +(66623*n^3-867526*n^2+3651197*n-4985254)*a(n-4) -20*(2*n-9)*(1027*n^2-13868*n+42561)*a(n-5) -10500*(n-5)*(2*n-9)*(2*n-11)*a(n-6)=0. - R. J. Mathar, Jul 27 2013

a(n) ~ 5^(2*n+3/2) / (9 * 4^n * n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Feb 03 2014