Combinatorics on bounded free Motzkin paths and its applications

TL;DR

This paper establishes bijections between various bounded Motzkin paths and core partitions, providing explicit formulas for counting specific types of core partitions with fixed corners.

Contribution

It introduces new bijections linking bounded free Motzkin paths, Dyck paths, and core partitions, enabling enumeration formulas for these combinatorial objects.

Findings

Bijection from bounded free Motzkin paths to bounded Motzkin prefixes

Explicit formulas for counting $t$-core and self-conjugate $t$-core partitions with fixed corners

Connections between bounded cornerless Motzkin paths and $t$-core partitions

Abstract

In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.