Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths (original) (raw)

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.

Dyck paths and restricted permutations

Discrete Applied Mathematics, 2006

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and S n (321) and S n (231), respectively. We also discuss permutations in S n avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete continuity between the Motzkin and the Catalan numbers.

Refinements of -Dyck paths

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.

Dyck paths with catastrophes modulo the positions of a given pattern

ArXiv, 2022

For any pattern p of length at most two, we provide generating functions and asymptotic approximations for the number of pequivalence classes of Dyck paths with catastrophes, where two paths of the same length are p-equivalent whenever the positions of the occurrences of the pattern p are the same.

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...

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.

Dyck Paths with Peak- and Valley-Avoiding Sets

Studies in Applied Mathematics, 2008

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences, which in turn induces an equivalence relation on avoiding sets.

An area-to-inv bijection between Dyck paths and 312-avoiding permutations

JOURNAL OF COMBINATORICS, 2001

The symmetric q, t-Catalan polynomial C n (q, t), which specializes to the Catalan polynomial C n (q) when t = 1, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund described statistics a(π) and b(π) on Dyck paths such that C n (q, t) = π q a(π) t b(π) where the sum is over all n × n Dyck paths. Specializing t = 1 gives the Catalan polynomial C n (q) defined by Carlitz and Riordan and further studied by Carlitz. Specializing both t = 1 and q = 1 gives the usual Catalan number C n . The Catalan number C n is known to count the number of n × n Dyck paths and the number of 312-avoiding permutations in S n , as well as at least 64 other combinatorial objects. In this paper, we define a bijection between Dyck paths and 312-avoiding permutations which takes the area statistic a(π) on Dyck paths to the inversion statistic on 312-avoiding permutations. The inversion statistic can be thought of as the number of (21) patterns in a permutation σ. We give a characterization for the number of (321), (4321), . . . , (k · · · 21) patterns that occur in σ in terms of the corresponding Dyck path.