# Counting Subwords Occurrences in Base-b Expansions

## TL;DR

This paper investigates the occurrence of subwords in base-b expansions of non-negative integers, establishing recurrence relations and asymptotic behavior for a related counting sequence using a tree structure.

## Contribution

It introduces a new approach using a tree structure to analyze subword counts in base-b expansions and proves the sequence's b-regularity.

## Key findings

- Derived recurrence relations for the sequence
- Proved the sequence's b-regularity
- Established asymptotic behavior of the summatory function

## Abstract

We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a generalization of the Pascal triangle to binomial coefficients of base-$b$ expansions. By using a convenient tree structure, we provide recurrence relations for $(S_b(n))_{n\ge 0}$ leading to the $b$-regularity of the latter sequence. Then we deduce the asymptotics of the summatory function of the sequence $(S_b(n))_{n\ge 0}$.