Markus Fulmek

Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters

We give bijective proofs for JacobiÂ?Trudi-type and Giambelli-type identities for symplectic and orthogonal characters. These proofs base on interpreting King and El-Sharkaways symplectic tableaux, Proctors odd and intermediate symplectic tableaux, Proctors and King and Welshs orthogonal tableaux, and Sundarams odd orthogonal tableaux in terms of certain families of nonintersecting lattice paths. This work is intended to be the counterpart of the GesselÂ?Viennot proof of the JacobiÂ?Trudi identities for Schur functions for the case of symplectic and orthogonal characters.

The Number of Rhombus Tilings of a Symmetric Hexagon which Contain a Fixed Rhombus on the Symmetry Axis, II

We compute the number of rhombus tilings of a hexagon with side lengths N, M,N , N, M, N, with N and M 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.

The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, I

We compute the number of rhombus tilings of a hexagon with sidesN,M,N, N,M,N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of lengthM.

A bijection between Proctor's and Sundaram's odd orthogonal tableaux

Abstract An explicit bijection between Proctors odd orthogonal tableaux and Sundarams odd orthogonal tableaux is given.

Dual rook polynomials

Abstract A theorem contained in the paper ‘A combinatoric formula’ by Wang, Lee and Tan (J. Math. Anal. Appl. 160 (1991) 500–503) gives rise to the definition of certain polynomials associated with boards: Stimulated by the analogy to the well-known rook polynomials, we call them ‘dual rook polynomials’. We show that in the cases of Ferrers boards and skew boards the evaluation of these polynomials at −1 always yields values −1, 0 or 1, generalizing the theorem cited above. Moreover, we evaluate these polynomials at −2. Finally, we state three conjectures that are quite well supported by empirical tests: Two of these conjectures are known to be true for rook polynomials.

On Pairs of Lattice Paths with a Given Number of Intersections

The purpose of this paper is to enumerate bijectively the number of ordered pairs of certain lattice paths: Consider paths in the integer lattice Z_Z that proceed with unit steps in either of the directions east or north. Let M k r, s be the set of ordered pairs of such paths that intersect exactly k times (i.e., that have k lattice points in common) in which both paths start at the origin (0, 0), the first path ends in (r, n&r) and the second path ends in (s, n&s). The intersection at the origin is not counted. Denote by M n, k r, s the cardinality of M n, k r, s : It is clear that M n, k r, s =M n, k s, r . A first formula for M n, k r, s was given in [1, Thm. 2]; the formula we want to prove here was given in [2, Thm. 5]. The definition for M k r, s in [1, 2] differs slightly from the one given above insofar as intersections at the endpoint (if r=s) are also not counted: This difference is irrelevant for the following theorem, but will simplify notation of the subsequent corollary.