Marko Thiel ; Nathan Williams - Strange Expectations and Simultaneous Cores

dmtcs:6364 - Discrete Mathematics & Theoretical Computer Science, April 22, 2020, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) - https://doi.org/10.46298/dmtcs.6364
Strange Expectations and Simultaneous CoresArticle

Authors: Marko Thiel 1; Nathan Williams ORCID2

  • 1 Institut für Mathematik [Zürich]
  • 2 Laboratoire de combinatoire et d'informatique mathématique [Montréal]

Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is (a2 −1)(b2 −1) 24, and showed that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is (a−1)(b−1)(a+b+1) 24. We apply P. Johnson's method to compute the variance and third moment. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of these formulae to simply-laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.


Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Published on: April 22, 2020
Imported on: July 4, 2016
Keywords: Combinatorics,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]

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