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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:
 5e+15 Combinatorial aspects of groups and algebras (MSC2010) 5e+10 Combinatorial aspects of representation theory
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##### References:
 [1] G.D. Carroll and D. Speyer, “The cube recurrence,” Electron. J. Combin. 11(1) (2004), 31 pp., Research Paper 73. · Zbl 1060.05004 [2] Fomin, S.; Zelevinsky, A., The Laurent phenomenon, Adv. in Appl. Math., 28, 119-144, (2002) · Zbl 1012.05012 [3] Fomin, S.; Zelevinsky, A.$$, Y$$-systems and generalized associahedra, Ann. of Math. (2), 158, 977-1018, (2003) · Zbl 1057.52003 [4] A. Henriques and J. Kamnitzer, “The octahedron recurrence and $${\sf gl}(n)$$ crystals,” to appear in Adv. in Math., available at Math.QA/0408114, 2005. [5] Mills, W. H.; Robbins, D. P.; Rumsey, H., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A, 34, 340-359, (1983) · Zbl 0516.05016 [6] J. Propp, “The many faces of alternating-sign matrices,” Discrete Math. Theor. Comput. Sci. Proc. AA (2001), 43-58. · Zbl 0990.05020 [7] J. Propp, Enumeration of Matchings: Problems and Progress, Volume 38 of New Perspectives in Algebraic Combinatorics. Cambridge Univ. Press, Cambridge, 1999, pp. 255-291. · Zbl 0937.05065 [8] Robbins, D. P.; Rumsey, H., Determinants and alternating sign matrices, Adv. in Math., 62, 169-184, (1986) · Zbl 0611.15008 [9] D.E. Speyer, “Perfect matchings and the octahedron recurrence,” to appear in J. Algebraic Combin. Math.CO/0402452, 2004.
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