More refined enumerations of alternating sign matrices
Abstract
We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_{n,i,j} that enumerate such matrices. We give a conjectural explicit formula for A_{n,i,j} and formulate several other conjectures about the sufficiency of the linear equations to determine the A_{n,i,j}'s and about an extension of the linear equations to the general d-refined enumerations.
Keywords
Cite
@article{arxiv.0903.5073,
title = {More refined enumerations of alternating sign matrices},
author = {Ilse Fischer and Dan Romik},
journal= {arXiv preprint arXiv:0903.5073},
year = {2009}
}
Comments
38 pages; added references and made minor changes to the introduction; source files now inlcude the Mathematica package RefinedASM