The sequence 1, 2, 7, 42, 429, 7436, ... counts the alternating sign matrices of order n. A conjectured formula for this sequence taunted combinatorialists for some fifteen years until it was proved first by Zeilberger and then by Kuperberg in 1996. Although the problem is combinatorial in nature, the method of Kuperberg's solution comes from statistical physics.

In my talk I will explain how the six-vertex model accounts for the residual entropy of ice. Then I will describe the connection to alternating sign matrices (ASM) and show the main idea behind the proof of the ASM conjecture. Finally, I will present an original theorem enumerating the states of the six-vertex model with certain non-standard boundary conditions. The formula was found at the Student Number Theory Seminar led by Eric Nathan Stucky and attended by Matěj Doležálek, Martin Raška, Ester Sgallová, and myself.