Proof of the Refined Alternating Sign Matrix Conjecture

TL;DR

This paper proves a stronger conjecture related to the enumeration of alternating sign matrices, building on previous work and employing advanced mathematical tools like orthogonal polynomials and q-calculus.

Contribution

It provides a proof for the refined enumeration conjecture of alternating sign matrices, extending prior results and utilizing sophisticated mathematical techniques.

Findings

Confirmed the refined enumeration formula for alternating sign matrices.

Extended the proof of the original conjecture to a more detailed case.

Utilized orthogonal polynomials and q-calculus in the proof.

Abstract

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the $r^{th}$ column, equals $A(n) {{n+r-2} \choose {n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and $q$-calculus, this stronger conjecture is proved.