Even-up words and their variants

TL;DR

This paper introduces and analyzes even-up and odd-up words over finite alphabets, deriving generating functions and exploring their combinatorial properties, with connections to well-known integer sequences like Motzkin, Riordan, and Catalan numbers.

Contribution

It defines new classes of words with digit-following constraints, derives explicit generating functions, and links these to classical combinatorial sequences, expanding understanding of their structure.

Findings

Derived explicit generating functions for eight classes of words.

Connected new word classes to Motzkin, Riordan, and Catalan numbers.

Provided combinatorial interpretations for various integer sequences.

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.