  In recent papers we have studied refined enumerations of alternating sign
matrices with respect to a fixed set of top and bottom rows. The present paper
is a first step towards extending these considerations to alternating sign
matrices where in addition a number of left and right columns are fixed. The
main result is a simple linear relation between the number of $n \times n$
alternating sign matrices where the top row as well as the left and the right
column is fixed and the number of $n \times n$ alternating sign matrices where
the two top rows and the bottom row is fixed. This may be seen as a first
indication for the fact that the refined enumerations of alternating sign
matrices with respect to a fixed set of top and bottom rows as well as left and
right columns can possibly be reduced to the refined enumerations where only a
number of top and bottom rows are fixed. For the latter numbers we provide a
system of linear equations that conjecturally determines them uniquely.
