This paper consists essentially of two parts. In the first one, we make a short introduction to traditional and modern Catalan cooking. In the second one, we provide a short list of my favorite restaurants.

Motivated by [13, Theorem 4.1], this paper extends Lewis’s bijection to a bijection between a more general class L(n, k, I) of permutations and the set of standard Young tableaux of shape 〈(k + 1)n〉, so the cardinality |L(n, k, I)| = f 〈(k+1)n〉, is independent of the choice of I ⊆ [n]. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end,...

For all positive integers n, we prove the following divisibility properties: (2n + 3) ( 2n n ) ∣∣∣∣3(6n 3n )( 3n n ) and (10n + 3) ( 3n n ) ∣∣∣∣21(15n 5n )( 5n n ) . This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product am ( am+bm−1 am )( an+bn an ) is divisible by m + n. In fact,...

This study examines the effects of lexical stress on vowel quality in Iberian Spanish and Central Catalan and the relationship between phonetic variation caused by stress and speech rate. For Catalan, both absence of stress and fast speech rate result in a shrunken vowel space. For Spanish, the effects of speech rate are in line with those found for Catalan. Yet, unstressed vowels are character...

We consider plane trees whose vertices are given labels from the set {1, 2, . . . , k} in such a way that the sum of the labels along any edge is at most k + 1; it turns out that the enumeration of these trees leads to a generalization of the Catalan numbers. We also provide bijections between this class of trees and (k + 1)-ary trees as well as generalized Dyck paths whose step sizes are k (up...

The study of quadrant marked mesh patterns in 132-avoiding permutations was initiated by Kitaev, Remmel and Tiefenbruck. We refine several results of Kitaev, Remmel and Tiefenbruck by giving new combinatorial interpretations for the coefficients that appear in the generating functions for the distribution of quadrant marked mesh patterns in 132-avoiding permutations. In particular, we study qua...

The delight of finding unexpected connections is one of the rewards of studying mathematics. In this talk, based on joint work with Ibtesam Bajunaid, Joel Cohen, and David Singman, I will discuss the connections that link the following seven superficially unrelated entities: (A) A function of the sort that calculus textbooks often use to show that a continuous function need not have a derivativ...

We prove that for any nonnegative integers n and r the binomial sum n ∑ k=−n ( 2n n− k ) k is divisible by 22n−min{α(n),α(r)}, where α(n) denotes the number of 1s in the binary expansion of n. This confirms a recent conjecture of Guo and Zeng [J. Number Theory, 130(2010), 172–186]. In 1976 Shapiro [3] introduced the Catalan triangle ( k n ( 2n n−k ) )n>k>1 and determined the sum of entries in t...

An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative, and combinatorial-recursive calculations of multidimensional integral expressions, which can describe partial solutions of these problems, are presented (the...

An interval vector is a (0, 1)-vector in Rn for which all the 1’s appear consecutively, and an interval-vector polytope is the convex hull of a set of interval vectors in Rn. We study three particular classes of interval vector polytopes which exhibit interesting geometric-combinatorial structures; e.g., one class has volumes equal to the Catalan numbers, whereas another class has face numbers ...