Two Combinatorial Interpretations of Rascal Numbers

TL;DR

This paper provides two combinatorial interpretations of Rascal Numbers, linking ascent counts in binary words and pattern avoidance in ascent sequences, enriching understanding of their combinatorial structure.

Contribution

It introduces two novel combinatorial interpretations of Rascal Numbers, connecting them to ascent statistics and pattern avoidance in sequences.

Findings

Rascal Numbers count ascents in binary words.

They relate to pattern avoidance in ascent sequences.

The interpretations connect to recent combinatorial research.

Abstract

The main goal of this paper is to assign two combinatorial interpretations to the elements of the Rascal Triangle defined by Angorro et al. The first interpretation involves counting ascents in binary words, while the second interpretation involves pattern avoidance in the ascent sequences studied by Duncan and Steingr\'imsson, and related directly to a more recent paper by Baxter and Pudwell.