Standard Young tableaux in a (2,1)-hook and Motzkin paths

The enumeration of standard Young tableaux (SYTs) is a fundamental problem in combinatorics and representation theory. While counting SYTs of bounded height k is known for k at most 5 with combinatorial proofs, much less is known for counting SYTs in a (k, l)-hook. In 2009 Regev enumerated standard Young tableaux of order n that are contained in a (2, 1)-hook. By a recurrence relation and the WZ method he proved that this number is 1/2(Sigma(j >= 1) ((n)(j))((n-j)(j))) + 1. In this paper we give a combinatorial proof of Regev's result by constructing a bijection between these tableaux and free Motzkin paths. (C) 2021 Elsevier B.V. All rights reserved.