用Motzkin数表示的斜形三列标准Young表的enumeration
On the enumeration of three-rowed standard Young tableaux of skew shape in terms of Motzkin numbers
摘要：
λ形标准杨氏表（SYT）的计数可以很容易地由钩长公式计算。1981年，Amitai Regev证明了最多有3行n元的SYT的个数等于第n个Motzkin数M_n。在2006年，Regev证明了在所有具有最多三个部分且具有n个条目的分区λ上的偏斜形状λ/（2，1）的SYT的总数是两个Motzkin数之差，M_{n-1} - M_{n-3}。Ekhad和Zeilberger使用计算机程序证明了Regev猜想。2009年，S. Eu发现了Motzkin路与最多三行的斜形SYT之间的一个双射，从而证明了Regev猜想，Eu还间接地证明了对于固定的μ =（μ1，μ2），所有最多三部分的划分λ上的斜形SYT的个数λ/μ可以表示为Motzkin数的一个线性组合.本文给出了一般情况下生成函数的一个显式公式：对于每个至多有三个部分的划分μ，生成函数给出了Motzkin数的线性组合的系数公式。我们还将证明这些生成函数与第二类切比雪夫多项式出乎意料地有关。
英文摘要：
The enumeration of standard Young tableaux (SYTs) of shape λ can be easily computed by the hook-length formula. In 1981, Amitai Regev proved that the number of SYTs having at most three rows with n entries equals the nth Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of skew shape λ/(2, 1) over all partitions λ having at most three parts 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 computer program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of skew shape with at most three rows to prove Regev's conjecture, and Eu also indirectly showed that for the fixed μ = (μ1,μ2) the number of SYTs of skew shape λ/μ over all partitions λ having at most three parts can be expressed as a linear combination of the Motzkin numbers. In this paper, we will find an explicit formula for the generating function for the general case: for each partition μ having at most three parts the generating function gives a formula for the coefficients of the linear combination of Motzkin numbers. We will also show that these generating functions are unexpectedly related to the Chebyshev polynomials of the second kind.
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