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SUMMARY:Michael Simkin (Harvard University)
DTSTART:20210930T141500Z
DTEND:20210930T160000Z
DTSTAMP:20260422T070030Z
UID:CJCS/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CJCS/31/">Th
 e number of $n$-queens configurations</a>\nby Michael Simkin (Harvard Univ
 ersity) as part of Copenhagen-Jerusalem Combinatorics Seminar\n\n\nAbstrac
 t\nThe $n$-queens problem is to determine $\\mathcal{Q}(n)$\, the number o
 f ways to place $n$ mutually non-threatening queens on an $n \\times n$ bo
 ard. We show that there exists a constant $\\alpha = 1.942 \\pm 3 \\times 
 10^{-3}$ such that $\\mathcal{Q}(n) = ((1 \\pm o(1))ne^{-\\alpha})^n$. The
  constant $\\alpha$ is characterized as the solution to a convex optimizat
 ion problem in $\\mathcal{P}([-1/2\,1/2]^2)$\, the space of Borel probabil
 ity measures on the square.\n\nThe chief innovation is the introduction of
  limit objects for $n$-queens configurations\, which we call \\textit{quee
 nons}. These are a convex set in $\\mathcal{P}([-1/2\,1/2]^2)$. We define 
 an entropy function that counts the number of $n$-queens configurations th
 at approximate a given queenon. The upper bound uses the entropy method. F
 or the lower bound we describe a randomized algorithm that constructs a co
 nfiguration near a prespecified queenon and whose entropy matches that fou
 nd in the upper bound. The enumeration of $n$-queens configurations is the
 n obtained by maximizing the (concave) entropy function in the space of qu
 eenons.\n\nBased on arXiv:2107.13460\n
LOCATION:https://researchseminars.org/talk/CJCS/31/
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