Finite Interpretation of the Hyper-Catalan Series Zero and its Powers

TL;DR

This paper provides a finite interpretation of the hyper-Catalan series zero and its powers by truncating the generating series to finite levels, offering combinatorial insights into their coefficients.

Contribution

It introduces a finite identity interpretation of the hyper-Catalan series zero at each truncation level and explores combinatorial derivations of powers of the series.

Findings

Finite identities at each truncation level for hyper-Catalan series zero.

Combinatorial derivations of coefficients for powers of the series.

Interpretation of series powers in terms of subdivided polygons.

Abstract

In 2025, Wildberger and Rubine showed the formal series zero of the univariate geometric polynomial is $\mathbf{S}$, the generating series for the hyper-Catalan numbers $\mathbf{C}_m$, which count the number of roofed subdivided polygons (subdigons) of type $\mathbf{m}$. We show that we can interpret this result as a finite identity at each level, where a level is a truncation of $\Sb$ to a given maximum number of vertices, edges, or faces (bounded by degree) of the associated subdigon types. We then explore powers $\mathbf{S}^r$, recounting Raney's and our own combinatorial derivations of its coefficients.