# Enumerating path diagrams in connection with $q$-tangent and $q$-secant numbers

## TL;DR

This paper develops a method to count specific height-restricted path diagrams linked to $q$-tangent and $q$-secant numbers using continued fractions and hypergeometric functions, extending previous unrestricted diagram results.

## Contribution

It introduces a novel enumeration approach for height-restricted path diagrams connected to $q$-tangent and $q$-secant numbers, generalizing prior work on unrestricted diagrams.

## Key findings

- Derived explicit formulas involving basic hypergeometric functions.
- Connected path diagram enumeration with continued fraction convergents.
- Extended enumeration results to restricted path diagrams.

## Abstract

We enumerate height-restricted path diagrams associated with $q$-tangent and $q$-secant numbers by considering convergents of continued fractions, leading to expressions involving basic hypergeometric functions. Our work generalises some results by M. Josuat-Verg\'es for unrestricted path diagrams [European Journal of Combinatorics 31 (2010) 1892].