# Generalized Pascal triangle for binomial coefficients of words

## TL;DR

This paper generalizes Pascal's triangle to finite words, exploring binomial coefficients as subsequence counts, and studies the resulting fractal-like structures modulo a prime, extending classical combinatorial and geometric concepts.

## Contribution

It introduces a novel generalization of Pascal's triangle for words and analyzes the associated fractal structures modulo a prime, linking combinatorics and geometry.

## Key findings

- Defined binomial coefficients for words as subsequence counts
- Described the limit set related to the generalized Pascal triangle modulo p
- Connected the structure to fractal geometry and Hausdorff distance

## Abstract

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpi\'nski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo $2$, we describe and study the first properties of the subset of $[0, 1] \times [0, 1]$ associated with this extended Pascal triangle modulo a prime $p$.