Hadamard products, lattice paths, and skew tableaux

This thesis concerns the computation of Hadamard products, the enumeration of lattice paths, and the enumeration of standard Young tableaux of skew shape.;The first chapter is about Hadamard products and tilings. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 11-ax-x2*1 1-bx-x2=1-x2 1-abx-&parl0;2+a2+b2&parr0;x2 -abx3+x4, where * denotes the Hadamard product. In a similar way, by considering tilings of a 2 x n rectangle with 1 x 1 and 1 x 2 bricks in the top row, and 1 x 1 and 1 x n bricks in the bottom row, we find an explicit formula for the Hadamard product 11-ax-x2*x m1-bx-xn.;The second chapter deals with lattice path enumerations using redundant generating functions. A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a surprising occurrence of Catalan numbers in counting paths that stay below the line y = 2x. In the second application we prove a conjecture of Niederhausen and Sullivan.;Finally, the third chapter is the enumeration of three-rowed standard Young tableaux of skew shape in terms of Motzkin numbers. The enumeration of standard Young tableaux (SYTs) of shape lambda 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 Mn. In 2006, Regev conjectured that the total number of SYTs of skew shape lambda/(2, 1) over all partitions lambda having at most three parts with n entries is the difference of two Motzkin numbers, Mn -1 - Mn-3. Ekhad and Zeilberger proved Regev's conjecture using a computer program. In his paper [3], 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 mu = (mu1, mu 2) the number of SYTs of skew shape lambda/mu over all partitions lambda having at most three parts can be expressed as a linear combination of Motzkin numbers. In this chapter, we will find an explicit formula for the generating function for the general case: for each partition mu 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.