Rhombus Tilings of a Hexagon with Three Fixed Border Tiles

Abstract

We compute the number of rhombus tilings of a hexagon with sides $a+2,b+2,c+2,a+2,b+2,c+2$ with three fixed tiles touching the border. The particular case $a=b=c$ solves a problem posed by Propp. Our result can also be viewed as the enumeration of plane partitions having $a+2$ rows and $b+2$ columns, with largest entry $\le c+2$, with a given number of entries $c+2$ in the first row, a given number of entries 0 in the last column and a given bottom-left entry.