The number of n-queens configurations

The n -queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. We show that there exists a constant & alpha; = 1.942 & PLUSMN; 3 x 10-3 such that Q(n) = ((1 & PLUSMN; o(1))ne-& alpha;)n. The constant & alpha; is characterized as the solution to a convex optimization problem in P([-1/2, 1/2]2), the space of Borel probability measures on the square. The chief innovation is the introduction of limit objects for n -queens configurations, which we call queenons. These form a convex set in P([-1/2, 1/2]2). We define an entropy function that counts the number of n -queens configurations that approximate a given queenon. The upper bound uses the entropy method of Radhakrishnan and Linial-Luria. For the lower bound we describe a randomized algorithm that constructs a configuration near a prespecified queenon and whose entropy matches that found in the upper bound. The enumeration of n -queens configurations is then obtained by maximizing the (concave) entropy function in the space of queenons. Along the way we prove a large deviations principle for n -queens configurations that can be used to study their typical structure.& COPY; 2023 Elsevier Inc. All rights reserved.