This material has been published in J. Combin. Theory Ser. A 100 (2002), 201-231, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Academic Press. This material may not be copied or reposted without explicit permission.

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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