Combinatorics on bounded free Motzkin paths and its applications (Q2699641)

Summary: 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.