  Monotone triangles are plane integer arrays of triangular shape with certain
monotonicity conditions along rows and diagonals. Their significance is mainly
due to the fact that they correspond to $n \times n$ alternating sign matrices
when prescribing $(1,2,...,n)$ as bottom row of the array. We define monotone
$(d,m)$--trapezoids as monotone triangles with $m$ rows where the $d-1$ top
rows are removed. (These objects are also equivalent to certain partial
alternating sign matrices.) It is known that the number of monotone triangles
with bottom row $(k_1,...,k_n)$ is given by a polynomial
$\alpha(n;k_1,...,k_n)$ in the $k_i$'s. The main purpose of this paper is to
show that the number of monotone $(d,m)$--trapezoids with prescribed top and
bottom row appears as a coefficient in the expansion of a specialization of
$\alpha(n;k_1,...,k_n)$ with respect to a certain polynomial basis. This
settles a generalization of a recent conjecture of Romik and the author. Among
other things, the result is used to express the number of monotone triangles
with bottom row $(1,2,...,i-1,i+1,...,j-1,j+1,...,n)$ (which is, by the
standard bijection, also the number of $n \times n$ alternating sign matrices
with given top two rows) in terms of the number of $n \times n$ alternating
sign matrices with prescribed top and bottom row, and, by a formula of
Stroganov for the latter numbers, to provide an explicit formula for the first
numbers. (A formula of this type was first derived by Karklinsky and Romik
using the relation of alternating sign matrices to the six--vertex model.)
