Osculating Paths and Oscillating Tableaux

Abstract

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points 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 paths correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they 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, the case of primary interest is tuples of paths with a fixed number l of vacancies and osculations. It is then shown that there exist natural bijections which map each such path tuple P to a pair (t, η), where η is an oscillating tableau of length l (i.e., a sequence of l+1 partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and t is a certain, compatible sequence of l weakly increasing positive integers. Furthermore, each vacancy or osculation of P corresponds to a partition in η whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for osculating paths involving sums over oscillating tableaux.

Multiply-refined enumeration of alternating sign matrices

“…ASMs with a fixed number of generalized inversions. Expressions for the coefficients of x p in Z n (x, 1), i.e., for the number of n × n ASMs with p generalized inversions, can be obtained using a result of Behrend [15,Cor. 14], and are given by Behrend (1, y), an expression for the coefficient of y m , i.e., for the number of n × n ASMs with m −1s, has been obtained by Cori, Duchon and Le Gac [53,108] , are bijections between ASM(n) and sets of monotone (or Gog) triangles with bottom row 1, .…”

On the weighted enumeration of alternating sign matrices and descending plane partitions

“…where the terms are written in an order which corresponds to that used in (5) and (6). It follows easily from the definitions of ASMs and DPPs that the generating functions satisfy…”

“…By using a bijection between ASMs and certain sets of osculating lattice paths, a bijection between DPPs and certain sets of nonintersecting lattice paths, and a result of Behrend [5,Cor. 14], which itself follows from a bijection [5,Thm. 13] between certain sets of paths and certain generalized oscillating tableaux, it can be shown that…”

“…Note that there is a closely-related bijection between ASMs and certain sets of osculating lattice paths. See, for example, Behrend [5, for further information and references. The statistical mechanical six-vertex or square ice model was itself first studied and solved by Lieb and Sutherland.…”

Higher Spin Alternating Sign Matrices

“…It can be seen, using (4) and (9), that a spin r/2 vertex model configuration corresponds to a semimagic square with line sum r if and only if each of its vertex…”