Letters of a given type in Catalan words: A continued fraction approach

Abstract

Generating functions related to Catalan words and frequencies of digits are obtained using continued fractions. This is fast, elegant, and flexible. It follows the philosophy of Philippe Flajolet from 1980.

Figures (3)

• Figure 1: Starting and ending at the bullet node. If $2n$ steps are done, the coefficient of $z^n$ is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$.
• Figure 2: Starting and ending at the bullet node. If $2n$ steps are done, the labels of the $n$ forward steps form a Catalan word. For $n=3$, these words are possible: $111$, $112$, $121$, $122$, $123$.
• Figure 3: Using variables to track forward letters (up steps).