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arXiv · 1508.05293

Strange Expectations

Marko Thiel·Nathan Williams
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Abstract

Let gcd(a,b)=1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a,b)-core is (a^2-1)(b^2-1)/24, and that this maximum was 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 extend P. Johnson's method to compute the variance to be ab(a-1)(b-1)(a+b)(a+b+1)/1440. By extending the definitions of "simultaneous cores" and "number of boxes" to affine Weyl groups, we give uniform generalizations of all three formulae above 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.

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Marko Thiel, Nathan Williams. 2015-08-21. Strange Expectations. https://arxiv.org/abs/1508.05293

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