A306292 Number of asymmetric Dyck paths of semilength n.

0, 0, 2, 8, 32, 112, 394, 1360, 4736, 16544, 58324, 207088, 741184, 2671008, 9688410, 35344800, 129620480, 477590080, 1767170812, 6563935664, 24465914304, 91481858208, 343058261572, 1289901443168

Offset

1,3

Comments

An asymmetric Dyck path is a path that generates a distinct Dyck path when traversed in opposite order.

Links

Table of n, a(n) for n=1..24.

J. Bonin, A. de Mier, and M. Noy, Lattice path matroids: enumerative aspects and Tutte polynomials, J Comb Theory Ser A 104 (2003), 63-94.

L.-H. Deng, E. Y. P. Deng, and L. W. Shapiro, The Riordan group and symmetric lattice paths, arXiv:0906.1844 [math.CO] (2009).

Zoe M. Himwich and Noah A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939.

Formula

a(n) = (2n)! / (n! (n+1))! - n! / ( (floor(n/2))! (ceiling(n/2))! ).

Example

For n=3, the a(2)=2 asymmetric Dyck paths are UUDDUD and UDUUDD.

Mathematica

Table[Binomial[2 n, n]/(n + 1) - Binomial[n, Floor[n/2]], {n, 0, 30}]

Prog

(PARI) a(n) = binomial(2*n, n)/(n+1) - binomial(n, n\2); \\ Michel Marcus, Jan 22 2020

Crossrefs

Equals twice A006079. Also A000108 minus A001405.

Sequence in context: A227971 A267661 A052481 * A067897 A145682 A363109

Adjacent sequences: A306289 A306290 A306291 * A306293 A306294 A306295

Keyword

nonn

Author

Noah A Rosenberg, Feb 04 2019

Status

approved