Six vertex model and enumerations of alternating-sign matrices

One more interaction between theoretical physics and mathematics will be discussed. There is a famous combinatorial problem to enumerate the so-called alternating-sign matrices, which are a generalization of the permutation matrices. This problem was solved by Doron Zeilberger in 1995. Much simpler solution was given by Greg Kuperberg in 1996. It was based on the one-to-one correspondence between the alternating-sign matrices and the states of the statistical six-vertex model. Here the problem is solved by calculation of the partition function, the main object of the statistical mechanics, for special values of the parameters. I will present an approach to the calculation of the partition function based on simple functional equations.