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Combinatorics of Generalized Bethe Equations

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A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over \({\mathbb{Z}^M}\), and on the other hand, they count integer points in certain M-dimensional polytopes.

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Correspondence to Evgeny K. Sklyanin.

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E. K. Sklyanin work supported by EPSRC Grant EP/H000054/1.

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Kozlowski, K.K., Sklyanin, E.K. Combinatorics of Generalized Bethe Equations. Lett Math Phys 103, 1047–1077 (2013).

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Mathematics Subject Classification (2010)