Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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
rate research
Read More
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.
Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions for $p(m)$ and $q(m)$ for all $m ge 1$. The two functions $p(n)$ and $q(n)$ can also be interpreted in terms of alternating paths among $n$ disjoint line segments in the plane, whose $2n$ endpoints are in convex position.
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.
In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions, recurrence relations and generating functions. We establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials of permutations in the symmetric group, which implies that the type B alternating Eulerian polynomials have gamma-vectors alternate in sign.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
