Strange Expectations and Simultaneous Cores

Marko Thiel (1) , Nathan Williams (2)

1 Institut für Mathematik [Zürich]
2 LaCIM - 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.

Mots clés: Combinatorics

Domaines: Combinatoire [math.CO]

Marko Thiel, Nathan Williams. Strange Expectations and Simultaneous Cores. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6364⟩. ⟨hal-02166326⟩