School Seminars and Colloquia

An Involution for Enumerating Osculating Lattice Paths and Alternating Sign Matrices


by Paul Fijn

Institution: MASCOS, University of Melbourne
Date: Wed 9th August 2006
Time: 3:15 PM
Location: AMSI Seminar Room, Ground Floor ICT Building, 111 Barry St

Abstract: Enumerative combinatorics is concerned with the establishment of "nice" formulae for the counting of various sets of objects. This talk is primarily concerned with paths on an integer lattice which can share vertices (but not edges) and must remain consistently ordered---osculating lattice paths.
It is known that there are many methods of enumerating two osculating lattice paths, and two methods for the problem of three paths. Unfortunately, none of these generalise to the N paths case. An outline of a proof for N paths is given, along with an introduction to basic enumerative techniques. Connections to other areas of statistical mechanics and combinatorics, such as Alternating Sign Matrices, the Bethe Ansatz and the 6-vertex model, will also be briefly discussed.

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