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Skew-standard tableaux with three rows. (English)
Adv. Appl. Math. 45, No. 4, 463-469 (2010).
Summary: Let $\cal T_3$ be the three-rowed strip. Recently {\it A. Regev} [“Probabilities in the $(k,l)$ hook,” Isr. J. Math. 169, 61‒88 (2009; Zbl 1200.05250)] conjectured that the number of standard Young tableaux with $n - 3$ entries in the “skew three-rowed strip” ${\mathcal{T}}_3 / (2,1,0) \text{ is } m_{n - 1} - m_{n - 3}$, a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were derived automatically and proved rigorously by Zeilberger via his powerful program and WZ method. It appears that each one is a linear combination of Motzkin numbers with constant coefficients. In this paper we will introduce a simple bijection between Motzkin paths and standard Young tableaux with at most three rows. With this bijection we answer Zeilberger’s question affirmatively that there is a uniform way to construct bijective proofs for all of those identities.