ABSTRACT:
Di Francesco proved that the number of configurations of a certain twenty-vertex model with fixed west boundary is equal to the number of domino tilings of the so-called Aztec triangle and conjectured that those numbers are counted by a product formula reminiscent of the famous Robbins numbers. Later, this conjecture was proved by Koutschan. Shortly thereafter, Fischer and Schreier-Aigner showed that a certain signed weighted enumeration of so-called arrowed Gelfand-Tsetlin patterns is also equal to the same product formula.
This leads to the question of whether there exists a direct combinatorial connection between these three classes of objects.
In this talk, we provide a probabilistic bijection between twenty-vertex configurations with free west boundary, generalizations of those with fixed west boundary, and the set of arrowed Gelfand-Tsetlin patterns with bounded entries, with no three equal entries in a row, and with no special little triangles. Since such arrowed Gelfand-Tsetlin patterns are the fixed-point set of arrowed Gelfand-Tsetlin patterns under a certain involution on arrowed Gelfand-Tsetlin patterns at the specialization giving the product formula, our map provides a combinatorial explanation of the equality of the numbers of the two aforementioned objects when the west boundary is fixed to a specific pattern.