The mysterious story of square ice, piles of cubes, and bijections
Ilse Fischera,1 and Matjazˇ Konvalinkab,c
aFaculty of Mathematics, University of Vienna, 1090 Vienna, Austria; bFaculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia; and cInstitute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia
Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved August 10, 2020 (received for review May 4, 2020) When combinatorialists discover two different types of objects that are counted by the same numbers, they usually want to prove this by constructing an explicit bijective correspondence. Such proofs frequently reveal many more details about the relation between the two types of objects than just equinumerosity. A famous set of problems that has resisted various attempts to find bijective proofs for almost 40 y is concerned with alternating sign matrices (which are equivalent to a well-known physics model for two- dimensional ice) and their relations to certain classes of plane partitions. In this paper we tell the story of how the bijections were found.
alternating sign matrices | plane partitions | bijective proofs
1. Introduction All mathematicians recognize 1, 1, 2, 3, 5, 8, 13, ... as Fibonacci numbers, and many of them, especially combinatorialists, recognize 1, 1, 2, 5, 14, 42, 132, ... as Catalan numbers. Both sequences appear frequently in combinatorics and other areas, they have many beautiful properties, and they are extremely well studied. However, there is the mysterious sequence 1, 1, 2, 7, 42, 429, 7,436, ..., whose terms are sometimes called Robbins numbers. They can be expressed with a product formula (Eq. 1), and the mystery comes from the fact that they count four different families of objects that, until now, could not be translated into one another. More precisely, there are many known objects enumerated by this sequence (13 of them are illustrated in different rows of Fig. 1). There are easy bijections between some of these objects, for example between alternating sign matrices and square ice configurations, and these bijections split the families of objects into four classes (separated by a line in Fig. 1): alternating sign matrices, descending plane partitions, totally symmetric self-complementary plane partitions, and alternating sign triangles. We invite readers looking for a challenge to try to guess the definitions of some of the objects in Fig. 1 that are not defined in this paper and to find bijections, for example, among the objects illustrated in the first six rows. However, until now, no bijection was known between any two of these four classes, despite the problem being open for almost four decades (for the first three classes; the fourth class was discovered only recently). In this paper, we describe such a bijection, one that connects alternating sign matrices and descending plane partitions. Let us emphasize right away that the bijection is far from simple. It is, however, completely explicit. We expect that the tools used in this paper can be used to find other complicated bijections. Let us mention that the combinatorial objects described are also commonly encountered in statistical mechanics, so this work has direct applications to questions in physics. 2. A Brief History An alternating sign matrix (ASM) is a square matrix with entries in {0, 1, −1} such that in each row and each column the nonzero entries alternate and sum up to 1. See row 1 of Fig. 1 for all ASMs of size 3 × 3. Robbins and Rumsey (1) introduced alternating sign matrices in the 1980s when studying their λ-determinant (a generalization of the classical determinant) and showing that the λ-determinant can be expressed as a sum over all alternating sign matrices of fixed size, thus generalizing the Leibniz formula that expresses the ordinary determinant as a sum over all permutations. Numerical experiments led Robbins and Rumsey (1) to conjecture that the number of n × n alternating sign matrices is given by the surprisingly simple product formula n−1 Y (3j + 1)! |ASMn | = . [1] (n + j )! j =0
Significance
In the early 1980s, it was discovered that alternating sign matrices (ASMs), which are also commonly encountered in statistical mechanics, are counted by the same numbers as two classes of plane partitions. Since then it has been a major open problem in this area to construct explicit bijections between the three classes of objects. In this paper we describe a bijective proof that relates ASMs to one of the two classes of plane partitions as well as a bijective proof of the fact that the numbers can be expressed by a simple product formula. The constructions are complicated, but they are completely explicit and also provided in the form of computer code.
Author contributions: I.F. and M.K. designed research, performed research, and wrote the paper.y The authors declare no competing interest.y This article is a PNAS Direct Submission.y Published under the PNAS license.y 1 To whom correspondence may be addressed. Email: ilse.fi[email protected] First published September 2, 2020.y
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MATHEMATICS They also conjectured a formula for the number of ASMs of size n × n with the unique 1 in the first row being in column i: