Abstract
Fermionic formulas in a combinatorial Bethe ansatz consist of sums of products of q-binomial coefficients. There exist refinements without a sum that are known to yield partition functions of box–ball systems with a prescribed soliton content. In this paper, such a refined fermionic formula is extended to the periodic box–ball system and a q-analog of the Bethe root counting formula for the XXZ chain at Δ = ∞.
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