Sen Peng Eu

Catalan and Motzkin numbers modulo 4 and 8

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.

Taylor expansions for Catalan and Motzkin numbers

In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae associated with Catalan and Motzkin numbers. In particular, our expansion for Catalan number provides a simple proof of the classic Chung?Feller theorem; similar result for the Motzkin paths with flaws is also given.

Refined Chung-Feller theorems for lattice paths

In this paper we prove a strengthening of the classical Chung-Feller theorem and a weighted version for Schroder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating schroder paths with flaws.

Area of Catalan paths on a checkerboard

It is known that the area of all Catalan paths of length n is equal to 4^n-2n+1n, which coincides with the number of inversions of all 321-avoiding permutations of length n+1. In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented.

Odd or even on plane trees

Over all plane trees with n edges, the total number of vertices with odd degree is twice the number of those with odd outdegree. Deutsch and Shapiro posed the problem of finding a direct two-to-one correspondence for this property. In this article, we give three different proofs via generating functions, an inductive proof and a two-to-one correspondence. Besides, we introduce two new sequences which enumerate plane trees according to the parity of the number of leaves. The explicit formulae for these sequences are given. As an application, the relation provides a simple proof for a problem concerning colored nets in Stanleys Catalan Addendum.

Standard Young tableaux and colored Motzkin paths

Abstract In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most 2 d + 1 rows and the set of SYTs with at most 2d rows.

Sign-balance identities of Adin-Roichman type on 321-avoiding alternating permutations

Abstract Adin and Roichman proved a set of refined sign-balance identities on 321-avoiding permutations respecting the last descent of the permutations, which we call the identities of Adin–Roichman type. In this work, we construct a new involution on plane trees that proves refined sign-balance properties on 321-avoiding alternating permutations respecting the first and last entries of the permutations respectively and obtain two sets of identities of Adin–Roichman type.

Constructions for cyclic sieving phenomena

We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs.

On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

A permutation σ ∈

Lattice paths and generalized cluster complexes

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