# Counting the number of non-zero coefficients in rows of generalized Pascal triangles

## TL;DR

This paper explores counting positive entries in generalized Pascal triangles based on words, establishing recurrence relations and connections with known sequences, and extends the approach to Zeckendorf numeration systems.

## Contribution

It introduces a tree-based method to derive recurrence relations for sequences counting positive entries in generalized Pascal triangles, linking them to known mathematical sequences and extending to Fibonacci-based systems.

## Key findings

- Established recurrence relations for the sequence of positive entries.
- Connected the sequence to the 2-regular Stern-Brocot sequence and Farey tree denominators.
- Extended the framework to Zeckendorf numeration systems and proved F-regularity.

## Abstract

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$ counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for $(S(n))_{n\ge 0}$. This leads to a connection with the $2$-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F-regularity of the corresponding sequence.