This material has been published in
J. Combin. Theory Ser. A
100 (2002), 201-231,
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Mihai Ciucu
and Christian Krattenthaler
Enumeration of lozenge tilings of hexagons with cut off corners
(23 pages)
Abstract.
Motivated by the enumeration of a class of plane partitions studied by
Proctor and
by considerations about symmetry classes of plane partitions, we consider
the problem of enumerating lozenge tilings of a hexagon with ``maximal
staircases''
removed from some of its vertices. The case of one vertex corresponds
to Proctor's
problem. For two vertices there are several cases to consider, and
most of them lead
to nice enumeration formulas. Two of these cases amount to evaluating
the determinant
$\det_{1\le i,j\le n}
(\binom{x+y+j}{x-i+2j}-\binom{x+y+j}{x+i+2j})$,
for which we prove a simple product formula that appears to be new.
For three or more vertices there do not seem to exist
nice product formulas in general, but in one special situation a lot
of factorization
occurs, and we pose the problem of finding a formula for the number of
tilings in this case.
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