Square-Ice Enumeration

Alain Lascoux

Séminaire lotharingien de combinatoire, Tome 42 (1998-1999)

Résumé

Starting with plane partitions possessing certain type of symmetries, many combinatorial objects came to the fore, the enumeration of which was the subject of intensive studies during the last twenty years, with of course, seminal contributions of George Andrews. Thanks to a detour through two-dimensional ice models, algebraic computations cristallised to the description of a certain determinant of Cauchy type. Dividing this determinant by some straightforward factors, one is reduced to studying a symmetric polynomial in two sets of variables. We show how to separate the variables with the help of divided differences, and obtain the desired symmetric function as a product of two rectangular matrices, each of them involving only one set of variables. In the same run, we reduce the dimension by 1 and factorize the determinant associated to the Bethe model of a 1-dimensional gas of bosons.