Pairs of noncrossing free Dyck paths and noncrossing partitions

Abstract

Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n + 1] with n + 1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm.

On three and four vicious walkers

“…In this section, we put the Labelle (1993) merging algorithm in a more general setting, and show that the direct correspondence formulated by Chen et al (2009) leads to a connection between pairs of converging walks and the classical ballot numbers.…”

“…Let (L 1 , L 2 ) be a 2-watermelon of length n, and let L 1 ¼ p 1 p 2 Á Á Á p n and L 2 ¼ q 1 q 2 Á Á Á q n , where p i , q i = U or D. Set U 0 = D and D 0 = U. Using the direct correspondence in Chen et al (2009), the watermelon (L 1 ,L 2 ) can be represented by a Dyck path of length 2n +2:…”

“…By reformulating the problem in terms of pairs of intersecting walks, we give a decomposition of a pair of converging walks, namely, two walks that do not intersect until they reach the same ending point, into two-chain watermelons, or 2-watermelons. Then we use Pó lya's formula for the number of 2-watermelons of length n to derive the formula for the number of 2-vicious walks of length n. In Section 4, we find a connection between the Labelle (1993) merging algorithm, in the form presented by Chen et al (2009), and the classical ballot numbers. In the last section, we give a reflection principle for the enumeration of configurations of 4-vicious walks with prescribed starting points.…”

A Reflection Principle for Three Vicious Walkers

“…In this section, we put the Labelle merging algorithm in a more general setting, and show that the direct correspondence formulated by Chen, Pang, Qu and Stanley [3] leads to a connection between pairs of converging walks and the classical ballot numbers.…”

“…By reformulating the problem in terms of pairs of intersecting walks, we give a decomposition of a pair of converging walks, that is, two walks that do not intersect until they reach the same ending point, into two-chain watermelons, or 2-watermelons. Then we can use Labelle's formula for the number of 2-watermelons of length n to derive the formula for the number of two vicious walks of length n. In the last section, we make a connection between pairs of converging walks and the classical ballot numbers, by applying the Labelle merging algorithm, in the form presented by Chen, Pang, Qu and Stanley [3],…”