Flattened Catalan Words

TL;DR

This paper introduces flattened Catalan words, a new combinatorial structure, and provides formulas, generating functions, and asymptotic analysis for their enumeration based on various run and valley parameters.

Contribution

It defines flattened Catalan words and derives explicit formulas, generating functions, and asymptotic expressions for their enumeration.

Findings

Derived formulas for flattened Catalan words based on run and valley parameters

Provided generating functions for various subclasses of flattened Catalan words

Established asymptotic expressions for counting flattened Catalan words

Abstract

In this work, we define flattened Catalan words as Catalan words whose runs of weak ascents have leading terms that appear in weakly increasing order. We provide generating functions, formulas, and asymptotic expressions for the number of flattened Catalan words based on the number of runs of ascents (descents), runs of weak ascents (descents), $\ell$-valleys, valleys, symmetric valleys, $\ell$-peaks, peaks, and symmetric peaks.