Statistics on Dyck paths (original) (raw)
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A Bijection on Bilateral Dyck Paths
It is known that both the number of Dyck paths with 2n steps and k peaks, and the number of Dyck paths with 2n steps and k steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly illustrates this equinumeracy. Moreover, we extend this bijection to bilateral Dyck paths. The restriction to Dyck paths preserves the number of contacts.
European Journal of Combinatorics, 2011
The classical Chung-Feller theorem tells us that the number of (n, m)-Dyck paths is the nth Catalan number and independent of m. In this paper, we consider refinements of (n, m)-Dyck paths by using four parameters, namely the peak, valley, double descent and double ascent. Let p n,m,k be the total number of (n, m)-Dyck paths with k peaks. First, we derive the reciprocity theorem for the polynomial P n,m (x) = ∑ n k=1 p n,m,k x k. In particular, we prove that the number of (n, m)-Dyck paths with k peaks is equal to the number of (n, n − m)-Dyck paths with n − k peaks. Then we find the Chung-Feller properties for the sum of p n,m,k and p n,m,n−k , i.e., the number of (n, m)-Dyck paths which have k or n − k peaks is 2(n+2) n(n−1) n k−1 n k+1 for 1 ≤ m ≤ n − 1 and independent of m. Finally, we provide a Chung-Feller type theorem for Dyck paths of semilength n with k double ascents: the total number of (n, m)-Dyck paths with k double ascents is equal to the total number of n-Dyck paths that have k double ascents and never pass below the x-axis, which is counted by the Narayana number. Let v n,m,k (resp. d n,m,k) be the total number of (n, m)-Dyck paths with k valleys (resp. double descents). Some similar results are derived.
New Formulas for Dyck Paths in a Rectangle
Combinatorics on Words, 2015
We consider the problem of counting the set of D a,b of Dyck paths inscribed in a rectangle of size a × b. They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers. By using Ferrers diagrams associated to Dyck paths, we derive formulas for the enumeration of D a,b with a and b non relatively prime, in terms of Catalan numbers.
Exterior Pairs and Up Step Statistics on Dyck Paths
The Electronic Journal of Combinatorics, 2011
Let mathcalCn\mathcal{C}_nmathcalCn be the set of Dyck paths of length nnn. 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 mathcalCn\mathcal{C}_nmathcalCn is equidistributed with the statistic 'number of up steps at height hhh with hequiv0h\equiv 0hequiv0 (mod 3)'. Moreover, for mge3m\ge 3mge3, we prove that the two statistics 'number of up steps at height hhh with hequiv0h\equiv 0hequiv0 (mod mmm)' and 'number of up steps at height hhh with hequivm−1h\equiv m-1hequivm−1 (mod mmm)' on the set mathcalCn\mathcal{C}_nmathcalCn are 'almost equidistributed'. Both results are proved combinatorially.
Reflnements of Dyck Paths with Flaws
The classical Chung-Feller theorem (2) tells us that the number of Dyck paths of length n with m ∞aws is the n-th Catalan number and independent on m. In this paper, we consider the reflnements of Dyck paths with ∞aws by four parameters, namely peak, valley, double descent and double ascent. Let pn;m;k be the number of all the Dyck paths of semi-length n with m ∞aws and k peaks. First, we derive the reciprocity theorem for the polynomial Pn;m(x) = n P k=1 p n;m;kxk. Then we flnd the Chung-Feller properties for the sum of pn;m;k and pn;m;n¡k. Finally, we provide a Chung-Feller type theorem for Dyck paths of length n with k double ascents: the number of all the Dyck paths of semi-length n with m ∞aws and k double ascents is equal to the number of all the Dyck paths that have semi-length n, k double ascents and never pass below the x-axis, which is counted by the Narayana number. Let vn;m;k (resp. dn;m;k) be the number of all the Dyck paths of semi-length n with m ∞aws and k valleys (r...
Enumeration of Dyck paths with air pockets
2022
We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for the subclass of non-decreasing Dyck paths with air pockets.
Counting Dyck Paths According to the Maximum Distance Between Peaks and Valleys
Journal of Integer Sequences
A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of up- steps u = (1, 1) and down-steps d = (1, 1) which never passes below the x-axis. Let Dn denote the set of Dyck paths of length 2n. A peak is an occurrence of ud (an upstep immediately followed by a downstep) within a Dyck path, while a valley is an occurrence of du. Here, we compute explicit formulas for the generating functions which count the members of Dn according to the maximum number of steps between any two peaks, any two valleys, or a peak and a valley. In addition, we provide closed expressions for the total value of the corresponding statistics taken over all of the members of Dn. Equivalent statistics on the set of 231-avoiding permutations of length n are also described.
Permutations and Pairs of Dyck Paths
ISRN Combinatorics, 2013
We define a mapvbetween the symmetric groupSnand the set of pairs of Dyck paths of semilengthn. We show that the mapvis injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.