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The enumeration of standard Young tableaux (SYTs) of shape {\lambda} can beeasily computed by the hook-length formula. In 1981, Amitai Regev proved thatthe number of SYTs having at most three rows with n entries equals the nthMotzkin number M_n. In 2006, Regev conjectured that the total number of SYTs ofskew shape {\lambda}/(2, 1) over all partitions {\lambda} having at most threeparts with n entries is the difference of two Motzkin numbers, M_{n-1} -M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computerprogram. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs ofskew shape with at most three rows to prove Regev's conjecture, and Eu alsoindirectly showed that for the fixed {\mu} = ({\mu}1,{\mu}2) the number of SYTsof skew shape {\lambda}/{\mu} over all partitions {\lambda} having at mostthree parts can be expressed as a linear combination of the Motzkin numbers. Inthis paper, we will find an explicit formula for the generating function forthe general case: for each partition {\mu} having at most three parts thegenerating function gives a formula for the coefficients of the linearcombination of Motzkin numbers. We will also show that these generatingfunctions are unexpectedly related to the Chebyshev polynomials of the secondkind.

On the enumeration of three-rowed standard Young tableaux of skew shape in terms of Motzkin numbers