Short proof of the ASM theorem avoiding the six-vertex model

摘要

Alternating sign matrix (ASM) counting is fascinating because it pushes the limits of counting tools. Nowadays, the standard method to attack such problems is the six-vertex model approach which involves computing a certain generating function of ASMs with-at first sight-nonorthodox weights originating from statistical mechanics. Still nobody has been able to use this technique to reprove the generalization of the ASM theorem that Zeilberger has actually established in the first proof of the ASM theorem, where he showed that there is the same number of n x k Gogtrapezoids as there is of n x k Magog-trapezoids nor has anybody proved Krattenthaler's conjectural generalization of this result. In 2007 I have presented a proof of the ASM theorem in a 12 page paper which does not involve the six-vertex model, but relies on another 19 page paper as well as Andrew's determinant evaluation that he used to enumerated descending plane partitions. Over the years I have discovered many simplifications of my original proof and it is the main purpose of this paper to present now a 9 page self-contained proof of the ASM theorem. In addition, I speculate on how to possibly transform this computational proof into a more combinatorial proof and I also provide a new constant term expression for the number of monotone triangles with prescribed bottom row. (C) 2016 Elsevier Inc. All rights reserved.