Even-up words and their variants

TL;DR

The paper studies parity-constrained words over a $k$-letter alphabet, introducing even-up, odd-up, and their weak and cyclic variants, and develops explicit generating functions for all eight classes. It extends the framework to Catalan words under similar restrictions and provides closed-form generating functions for the resulting families, with special cases tied to Motzkin numbers, Riordan numbers, and generalized Catalan numbers. Methodologically, the authors employ linear-algebra formulations, creative telescoping, and enumeration techniques to obtain exact expressions. The work yields new combinatorial interpretations for several classical integer sequences and broadens the tabulation of restricted-word classes within enumerative combinatorics.

Abstract

Inspired by OEIS sequence A377912, which consists of the nonnegative integers in which every even digit (except possibly the last) is immediately followed by a strictly larger digit, we define even-up and odd-up words over an alphabet of size~$k$ via similar constraints. We introduce and analyze weak and cyclic variants of these words, deriving explicit generating functions for all eight resulting classes. We then study Catalan words under analogous restrictions. Our results provide new combinatorial interpretations for many integer sequences, including the Motzkin numbers, the Riordan numbers, and the generalized Catalan numbers.

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8