scholarly journals Proof of the Alternating Sign Matrix Conjecture

10.37236/1271 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Alternating Sign Matrix ◽  
Sign Matrix

The number of $n \times n$ matrices whose entries are either $-1$, $0$, or $1$, whose row- and column- sums are all $1$, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $$[1!4! \dots (3n-2)!] \over [n!(n+1)! \dots (2n-1)!],$$ as conjectured by Mills, Robbins, and Rumsey.

scholarly journals Higher Spin Alternating Sign Matrices

10.37236/1001 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Roger E. Behrend ◽  
Vincent A. Knight
Keyword(s):  
Convex Polytope ◽  
Ehrhart Polynomial ◽  
Alternating Sign Matrices ◽  
Integer Points ◽  
Wall Boundary Conditions ◽  
Alternating Sign Matrix ◽  
Higher Spin ◽  
Statistical Mechanical ◽  
Sign Matrix ◽  
Integral Convex Polytope

We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer $r$, all complete row and column sums are $r$, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin $r/2$ statistical mechanical vertex models with domain-wall boundary conditions. The case $r=1$ gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size $n$ are the integer points of the $r$-th dilate of an integral convex polytope of dimension $(n{-}1)^2$ whose vertices are the standard alternating sign matrices of size $n$. It then follows that, for fixed $n$, these matrices are enumerated by an Ehrhart polynomial in $r$.

scholarly journals On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope

2019 ◽  
Vol 62 (1) ◽  
pp. 128-163 ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Jessica Striker
Keyword(s):  
Flow Polytopes ◽  
Alternating Sign Matrix ◽  
Sign Matrix

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

2001 ◽  
Vol 108 (1) ◽  
pp. 85 ◽  
Author(s):  
Doron Zeilberger ◽  
David M. Bressoud
Keyword(s):  
Alternating Sign Matrix ◽  
Sign Matrix

scholarly journals The Alternating Sign Matrix Polytope

10.37236/130 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Convex Hull ◽  
Complete Characterization ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Alternating Sign Matrix ◽  
Sign Matrix ◽  
Equivalent Description

We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.

scholarly journals De Finetti Lattices and Magog Triangles

10.37236/9246 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Andrew Beveridge ◽  
Ian Calaway ◽  
Kristin Heysse
Keyword(s):  
Boolean Lattice ◽  
Order Ideal ◽  
Strict Sense ◽  
Surprising Result ◽  
Linear Extensions ◽  
Alternating Sign Matrix ◽  
Ballot Numbers ◽  
Proof Techniques ◽  
Sign Matrix ◽  
Standard Young Tableau

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of  shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$  of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

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