A periodicity theorem for the octahedron recurrence

A periodicity theorem for the octahedron recurrence

The octahedron recurrence lives on a 3-dimensional lattice and is given by f (x, y, t + 1) = ( f (x + 1, y, t) f (x − 1, y, t) + f (x, y + 1, t) f (x, y − 1, t))/ f (x, y, t − 1). In this paper, we investigate a variant of this recurrence which lives in a lattice contained in [0, m] × [0, n] × R. Following Speyer, we give an explicit non-recursive formula for the values of this recurrence and u...

Arrays and the Octahedron Recurrence

The Octahedron Recurrence and Rsk-correspondence

Wemake the statement rigorous that the Robinson–Schensted–Knuth correspondence is a tropicalization of the Dodgson condensation rule.

Perfect matchings and the octahedron recurrence

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the L...

Bessenrodt-Stanley Polynomials and the Octahedron Recurrence

We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data. This leads to generalizations and explicit expressions as path or dimer partition functions.