Plane partitions I: a generalization of MacMahon's formula

Abstract

The number of plane partitions contained in a given box was shown by MacMahon to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of side-lengths $a,b,c,a,b,c$ (in cyclic order) and angles of 120 degrees. We present a generalization in the case $b=c$ by giving simple product formulas enumerating lozenge tilings of regions obtained from a hexagon of side-lengths $a,b+k,b,a+k,b,b+k$ (where $k$ is an arbitrary non-negative integer) and angles of 120 degrees by removing certain triangular regions along its symmetry axis.

arXiv comment35 pages, 34 figures. New to this version: a few typos were corrected, and the journal information is included. Memoirs of Amer. Math. Soc., accepted, to appear

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