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Abstract: We study a further refinement of the standard refined enumeration ofalternating sign matrices ASMs according to their first two rows instead ofjust the first row, and more general -d-refined- enumerations of ASMs accordingto the first d rows. For the doubly-refined case of d=2, we derive a system oflinear equations satisfied by the doubly-refined enumeration numbers A {n,i,j}that enumerate such matrices. We give a conjectural explicit formula forA {n,i,j} and formulate several other conjectures about the sufficiency of thelinear equations to determine the A {n,i,j}-s and about an extension of thelinear equations to the general d-refined enumerations.



Author: Ilse Fischer, Dan Romik

Source: https://arxiv.org/







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