Implicit Divided Differences, Little Schr\"oder Numbers, and Catalan Numbers

TL;DR

This paper explores the combinatorial structures behind a formula for divided differences of implicitly defined functions, revealing connections to Little Schröder and Catalan numbers through six equivalent characterizations.

Contribution

It provides a detailed combinatorial analysis of the sequence of terms in a formula for divided differences, linking it to well-known combinatorial numbers and characterizations.

Findings

Sequence of terms relates to Little Schröder and Catalan numbers

Six equivalent characterizations of the sequence _n are established

Deepens understanding of the combinatorial structure in implicit differentiation

Abstract

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of $y$ expressed in terms of bivariate divided differences of $g$, by analyzing the number of terms $a_n$ in this formula. The main result describes six equivalent characterizations of the sequence $\{a_n\}$.