An Algebraic Interpretation of the Super Catalan Numbers

TL;DR

This paper develops an algebraic integration framework over circles in Euclidean geometry over fields of characteristic zero, providing an algebraic interpretation of super Catalan numbers through this theory.

Contribution

It introduces a novel algebraic integration theory over circles in Euclidean geometry over arbitrary fields, linking super Catalan numbers to algebraic integrals of monomials.

Findings

Established an algebraic integration functional invariant under SO(2, F)

Connected super Catalan numbers to algebraic integrals of specific monomials

Extended polynomial integration concepts to general fields of characteristic zero

Abstract

We extend the notion of polynomial integration over an arbitrary circle $C$ in the Euclidean geometry over general fields $\mathbb F$ of characteristic zero as a normalized $\mathbb F$-linear functional on $\mathbb{F}\left[\alpha_1, \alpha_2\right]$ that takes polynomials that evaluate to zero on $C$ to zero and is $\mathrm{SO}(2,\mathbb{F})$-invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!}$$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha_1^{2m}\alpha_2^{2n}$.