Related papers: Counting $r\times s$ rectangles in (Catalan) words
The rectangle capacity, a word statistic that was recently introduced by the author and Mansour, counts, for two fixed positive integers $r$ and $s$, the number of occurrences of a rectangle of size $r\times s$ in the bargraph…
Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…
We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation…
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last…
Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan…
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the…
The distributions of the number of occurrences of words (the distributions of words for short) play key roles in information theory, statistics, probability theory, ergodic theory, computer science, and DNA analysis. Bassino et al. 2010 and…
This work is a continuation of some recent articles presenting enumerative results for Catalan words avoiding one or a pair of consecutive or classical patterns of length $3$. More precisely, we provide systematically the bivariate…
A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms. These objects are enumerated by the well-known Catalan numbers, and have been introduced with the aim of giving a common language to most of the structures…
We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular,…
We introduce a new bargraph statistic that we call black-white cell capacity. It is obtained by coloring the cells of the bargraph in a chessboard style and recording the numbers of black and white cells contained in the bargraph. We study…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
A Catalan word is a sequence $w_1w_2\cdots w_n$ of nonnegative integers such that $w_1=0$ and $w_{i}\leq w_{i-1}+1$ for $2\leq i\leq n$. Given a Catalan word, we construct a column-convex polyomino (or \emph{bargraph}) by placing, at…
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
The $p$-th power of the logarithm of the Catalan generating function is computed using the Stirling cycle numbers. Instead of Stirling numbers, one may write this generating function in terms of higher order harmonic numbers.
Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $r \times s$ rectangle. Our main…
In this paper, we study additional aspects of the capacity distribution on the set $\mathcal{B}_n$ of compositions of $n$ consisting of $1$'s and $2$'s. Among our results are further recurrences for this distribution as well as formulas for…
A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…