Linear relations of refined enumerations of alternating sign matrices

Details Linear relations of refined enumerations of alternating sign matrices Linear relations of refined enumerations of alternating sign matrices

2010-08-03

arxiv.org/abs/1008.0527v1

Mathematics

Combinatorics

Scientific paper

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.

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