Humps in Motzkin paths and standard Young tableaux in a $(2,1)$-hook

We calculate the number of humps and peaks in Motzkin paths with a given height, and calculate the number of standard Young tableaux (SYTs) in a $(2,1)$-hook with the difference of the first two parts fixed, which refine Regev’s results in 2009. We also give new combinatorial proofs of Regev’s results, and reveal some new recurrence relations related to humps, free Motzkin paths and SYTs.