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Register a new userExterior Pairs and Up Step Statistics on Dyck Paths
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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.
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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 consecutive valleys (from left to right) is at least $d$, we say that $P$ is a restricted $d$-Dyck path. The emph{area} of a Dyck path is the sum of the absolute values of $y$-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of $d$-Dyck paths. We give a bivariate generating function to count the number of the $d$-Dyck paths with respect to the the semi-length and number of peaks. After that, we analyze in detail the case $d=-1$. Among other things, we give both, the generating function and a recursive relation for the total area.
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 $(1,-1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: number of $u$-segments, number of internal $u$-segments and number of $(u,h)$-segments. The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to $u$-segments and $(u,h)$-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.
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. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q=1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.
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 done by a simple walking algorithm on a $Tin mathcal{T}_n^k$. We find their idea naturally extends for $mathbf{k}^pm$-Dyck paths, and also for $mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $mathcal{T}_mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.
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 $n$, consisting of diagonal up- and down-steps and being confined to a strip of bounded width, to negative $n$ produces numbers of alternating sequences of integers that are bounded from below and from above. We show that this reciprocity extends to families of non-intersecting bounded up-down paths and certain arrays of alternating sequences which we call alternating tableaux. We provide as well weight
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