Related papers: Connectivity of Alternating Sign Triangle
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are…
Alternating sign triangles have been introduced by Ayyer, Behrend and Fischer in 2016 and it was proven that there is the same number of alternating sign triangles with $n$ rows as there is of $n\times n$ alternating sign matrices. Later on…
I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone…
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe.…
We show that there is the same number of (n,l)-alternating sign trapezoids as there is of column strict shifted plane partitions of class l-1 with at most n parts in the top row, thereby proving a result that was conjectured independently…
This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently…
Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend and the author, and it was proven that there is the same number of ASTs with n rows as there is of nxn alternating sign matrices (ASMs). We prove a conjecture…
In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative…
Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but no bijective proof for any of these equivalences has been…
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement…
Lascoux and Sch\"utzenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices…
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices,…
In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…
We define the alternating sign matrix polytope as the convex hull of nxn 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…
Since the 1980s, it has been known that essential surfaces in alternating link complements can be isotoped to be transverse to the link diagram almost everywhere, with the exception of some well-understood intersections, and described…
For a spanning tree T of a connected graph G and for a labelling \phi: E(T) \rightarrow {+, -}, \phi is called an alternating sign on a spanning tree T of a graph G if for any cotree edge e \in E(G)-E(T), the unique path in T joining both…
Since the alternating sign matrix conjecture, proposed by Mills, Robbins, and Rumsey in 1982, was proved by Zeilberger and Kuperberg, several refined enumerations have been considered. In particular, Behrend et al. obtained a quadruply…
An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em…
We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we…
This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for…