Counting Humps in Motzkin paths

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.Comment: 8 pages, 2 Figure

Motzkin paths, Motzkin polynomials and recurrence relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce sonic properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof

Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges\u27s Theorem

In a companion article dedicated to the enumeration aspects, we showed how to obtain closed form formulas for the generating functions of walks, bridges, meanders, and excursions avoiding any fixed word (a pattern p). The autocorrelation polynomial of this forbidden pattern p (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) plays a crucial role. In this article, we get the asymptotics of these walks. We also introduce a trivariate generating function (length, final altitude, number of occurrences of p), for which we derive a closed form. We prove that the number of occurrences of p is normally distributed: This is what Flajolet and Sedgewick call an instance of Borges\u27s theorem. We thus extend and refine the study by Banderier and Flajolet in 2002 on lattice paths, and we unify several dozens of articles which investigated patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. Our approach relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. The situation is much more involved than in the Banderier-Flajolet work: forbidden patterns lead to a wider zoology of asymptotic behaviours, and we classify them according to the geometry of a Newton polygon associated with these constrained walks, and we analyse what are the universal phenomena common to all these models of lattice paths avoiding a pattern

On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution

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Distribution of peak heights modulo kk and double descents on kk-Dyck paths

We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these statistics, together with the number of double descents, are jointly equidistributed with any of their permutations. We also generalize this result to generalized Motzkin paths and generalized ballot paths.Comment: 11 pages, 3 figure