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We compute the number of rhombus tilings of a hexagon with side lengths having the same parity, which contain a particular rhombus next to the center of the hexagon. The special case N = M of one of our results solves a problem posed by Propp. In the proofs, Hankel determinants featuring Bernoulli numbers play an important role. 1. Introduction Let a, b and… (More)

We give bijective proofs for Jacobi{Trudi-type and Giambelli-type identities for symplectic and orthogonal characters. These proofs base on interpreting King and El-Sharkaway's symplectic tableaux, Proctor's odd and intermediate symplectic tableaux, Proctor's and King and Welsh's orthogonal tableaux, and Sundaram's odd orthogonal tableaux in terms of… (More)

An explicit bijection between Proctor's odd orthogonal tableaux and Sundaram's odd orthogonal tableaux is given.

We present a " method " for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi–Trudi identity. We illustrate this " method " by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for… (More)

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström–Gessel–Viennot-method and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides " graphical proofs " for classical deter-minantal identities like the… (More)

The purpose of this note is to exhibit clearly how the " graphical condensation " identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn–Percus method for the enumeration of matchings. Knuth termed the relevant identities " overlapping Pfaffian " identities and the key concept of proof " su-perpositions of matchings… (More)

In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an alge-braic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a " purely combinatorial " proof for this identity; i.e., a proof involving only combinatorial arguments. Since these arguments eventually employ a generalization of Laplace's… (More)

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length n) to the case of p– watermelons with a wall (i.e., to a certain family of p nonintersecting Dyck paths; simple Dyck paths being the special case p = 1.) We work out this asymptotics for the case p = 2 only, since the… (More)