# Osculating Paths and Oscillating Tableaux

*Statistical Mechanics/Combinatorics Seminar*

*by Dr. Roger E. Behrend*

*Institution:*School of Mathematics, Cardiff University

*Date: Thu 6th September 2007*

*Time: 3:15 PM*

*Location: Room 213, Richard Berry Building, University of Melbourne*

*Abstract*: In this talk, the combinatorics of certain osculating lattice paths

will be discussed, and a natural relationship with oscillating tableaux will

be obtained. More specifically, path tuples will be considered in which each path has a fixed start and end point on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples include cases which correspond to alternating sign matrices and

various subclasses thereof. Referring to points of the rectangle through which no or two paths pass as vacancies or osculations respectively, it will be shown that there exist bijections which map each path tuple P with l vacancies and osculations to a pair (t, eta), where eta is an oscillating tableau of length l (i.e., a sequence of l plus 1 Young diagrams,

starting with the empty diagram, in which successive diagrams differ by a

single square), and t is a certain, compatible sequence of l weakly-increasing positive integers. In these bijections, which can be

regarded as generalizations of well-known bijections between certain tuples of nonintersecting lattice paths and semistandard Young tableaux, each vacancy or osculation of P corresponds to a Young diagram in |eta

which is obtained from its predecessor by respectively the addition or deletion of a square.

*For More Information:* Dr. Iwan Jensen I.Jensen@ms.unimelb.edu.au