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Definition & Example
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- An **alternating sign matrix (ASM) of order $n$** is an $n \times n$-matrix with entries $-1$, $0$ or $+1$ such that the sum of the entries in each row and each column equals $1$ and the nonzero entries alternate in sign along each row and each column.
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- There are $\displaystyle M(n):=\prod_{i=0}^{n-1}\frac{(3i+1)!}{(n+i)!}$ ASMs of order $n$, see [A005130](https://oeis.org/A005130).
Properties
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- Every permutation matrix is an ASM.
- There is exactly one $1$ in the first row of an ASM. Zeilberger <> proved that the number of ASMs of order $n$ whose unique $1$ of the first row is in the $r$th column is given by $\displaystyle \frac{\binom{n+r-2}{n-1} \binom{2n-1-r}{n-1}}{\binom{3n-2}{n-1}} \prod_{i=0}^{n-1}\frac{(3i+1)!}{(n+i)!}$. See <> for a more refined enumeration of ASMs.
- There are eight symmetry classes of ASMs. Except for the cases of diagonally symmetric ASMs, totally symmetric ASMs, and diagonally and antidiagonally symmetric ASMs of even order, there are proven product formulae for the straight enumeration in all classes, see <>, [](https://arxiv.org/abs/math/0008045)<>, <> .
- There is a bijective correspondence between ASMs and **monotone triangles**. Monotone triangles of order $n$ are [Gelfand-Tsetlin patterns](/GelfandTsetlinPatterns) with strictly decreasing rows and top row $n,n-1,\dots,1$.
- ASMs are in bijection with **fully-packed loop configurations (FPL)**. The enumeration problem of FPLs with a given link pattern has only been solved for certain link patterns. Razumov and Stroganov conjectured a relation between the enumeration of FPLs with a given link pattern and the dense $O(1)$-loop model. It was proved combinatorially by Cantini and Sportiello <>.
- There is a bijective correpsondence between ASMs and the **six-vertex model** with domain wall boundary conditions.
- Domino tilings of **Aztec diamonds** correspond to the so-called *$2$-enumeration* of ASMs <>, <>.
Remarks
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- The notion of ASMs arose in Robbins' and Rumsey's study of **$\lambda$-determinants** in the early 1980s <>. They conjectured with Mills the enumeration formula for ASMs <>, which had been known as the *alternating sign matrix conjecture* before being independently proved to hold true by Zeilberger <>, Kuperberg <>, and Fischer <>. See <> for more information about the history of the alternating sign matrix conjecture.
- $M(n)$ enumerates several other objects: **descending plane partitions** whose parts do not exceed $n$ <>, <>; **totally symmetric self-complementary plane partitions** inside a $2n \times 2n \times 2n$-box <>, <>; **alternating sign triangles** with $n$ rows [](https://arxiv.org/abs/1611.03823)<>. However, no bijections between these objects and ASMs have been found so far.
References
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Sage examples
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{{{#!sagecell
for n in [2,3,4]:
ASMs = AlternatingSignMatrices(n)
print n, ASMs.cardinality()
for ASM in AlternatingSignMatrices(3):
print ASM
print
}}}
Technical information for database usage
========================================
- An alternating sign matrix is uniquely represented as a **list of lists** representing its rows.
- Alternating sign matrices are **graded by its size**.
- The database contains all alternating sign matrices of size at most 6.