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A periodicity theorem for the octahedron recurrence. (English) Zbl 1125.05106
Summary: 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 \([0,m] \times [0,n] \times \mathbb R\). Following D. E. Speyer [J. Algebr. Comb. 25, 309–348 (2007; Zbl 1119.05092)], we give an explicit non-recursive formula for the values of this recurrence and use it to prove that it is periodic of period \(n + m\). We then proceed to show various other hidden symmetries satisfied by this bounded octahedron recurrence.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E10 Combinatorial aspects of representation theory
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