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Séminaire Lotharingien de Combinatoire, B42p (1999), 15 pp.

# Alain Lascoux

#
Square-Ice Enumeration

**Abstract.**
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.

Received: December 16, 1998; Accepted: May 26, 1999.

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