# Determinants and alternating sign matrices

```@article{Robbins1986DeterminantsAA,
title={Determinants and alternating sign matrices},
author={David P. Robbins and Howard Rumsey},
year={1986},
volume={62},
pages={169-184}
}```
• Published 1 November 1986
• Mathematics
Let M be an n by n matrix. By a connected minor of M of size k we mean a minor formed from k consecutive rows and k consecutive columns. We give formulas for det M in terms of connected minors, one involving minors of two consecutive sizes and one involving minors of three consecutive sizes. The formulas express det M as sums indexed by sets of alternating sign matrices. These matrices are described here and by W. H. Mills, D. P. Robbins, and H. Rumsey, Jr. (Invent. Math.66 (1982), 73–87; J… Expand
162 Citations
On the weighted enumeration of alternating sign matrices and descending plane partitions
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2012
It is proved that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1@?s and m+p inversions is equal to the number to descending plane partitions (DPPs). Expand
An Alternating Sum of Alternating Sign Matrices
An alternating-sign matrix (ASM) is a square matrix with entries from {-1, 0,1}, row and column sums of 1, and in which the nonzero entries in each row and column alternate in sign. ASMs have manyExpand
Enumeration of symmetry classes of alternating sign matrices and characters of classical groups
An alternating sign matrix is a square matrix with entries 1, 0 and −1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along eachExpand
Symmetry Classes of Alternating Sign Matrices
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate inExpand
Diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Mathematics, Physics
• 2017
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are inExpand
A new determinant for the Q-enumeration of alternating sign matrices
• F. Aigner
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2021
It is proved that her formula, when replacing the third root of unity by an indeterminate \$q\$, is actually the \$(2+q+q^{-1})\$-enumeration of alternating sign matrices. Expand
A directed graph structure of alternating sign matrices
Abstract We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet).Expand
A ug 2 00 0 Symmetry Classes of Alternating Sign Matrices
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, −1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate inExpand
Alternating sign trapezoids and a constant term approach
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 aExpand
A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux
It is proved the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally asymmetric plane partitions. Expand

#### References

SHOWING 1-8 OF 8 REFERENCES
Alternating Sign Matrices and Descending Plane Partitions
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 1983
This paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented. Expand
Proof of the Macdonald conjecture
• Mathematics
• 1982
A plane partition of n is an array ~=(a i ) , not necessarily square, of positive integers, which has non-increasing rows and columns, such that the sum of its elements is n. With such an array weExpand
Plane partitions (III): The weak Macdonald conjecture
Thus the plane partitions of 3 are 3, 21, 2, 111, 11, 1 1 1 1 1. There are some well-known theorems and open questions related to plane partitions in which the number of rows and columns and the sizeExpand
Symmetric functions and Hall polynomials
I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functionsExpand
DWYER, “Linear Computations,
• Wiley. New York,
• 1951