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Let $C_n$ be the set of Dyck paths of length $n$. In this paper, by a new automorphism of ordered trees, we prove that the statistic `number of exterior pairs, introduced by A. Denise and R. Simion, on the set $C_n$ is equidistributed with the statistic `number of up steps at height $h$ with $hequiv 0$ (mod 3). Moreover, for $mge 3$, we prove that the two statistics `number of up steps at height $h$ with $hequiv 0$ (mod $m$) and `number of up steps at height $h$ with $hequiv m-1$ (mod $m$) on the set $C_n$ are `almost equidistributed. Both results are proved combinatorially.
In this paper we study a subfamily of a classic lattice path, the emph{Dyck paths}, called emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of two consecu
A {em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $kgeq 0$, up-steps $(1,1)$, and down-steps $(1,-
We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable diagonal inversion statistic.
Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=knpm 1$. They introduced an intermediate family $mathcal{T}_n^k$ of certain standard Young tableau. Then inverting the sweep map is
The theme of this article is a reciprocity between bounded up-down paths and bounded alternating sequences. Roughly speaking, this ``reciprocity manifests itself by the fact that the extension of the sequence of numbers of paths of length $n$, consis