Implicit Divided Differences, Little Schröder Numbers, and Catalan Numbers
Provides a combinatorial interpretation and multiple characterizations of a sequence arising in numerical analysis, but the result is incremental and of interest primarily to specialists in combinatorial analysis.
The paper studies the combinatorial structure of a formula for divided differences of implicitly defined functions, deriving six equivalent characterizations of the sequence counting terms in the formula, which turns out to be the little Schröder numbers.
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 $\{a_n\}$.