奇怪的期望
Strange Expectations
摘要：
设gcd（a，b）=1. J. Olsson和D.斯坦顿证明了在一个同时存在的（a，b）-核中，盒子的最大数量是（a^2-1）（b^2-1）/24，并且这个最大值是由一个唯一的核实现的。P. Johnson将Ehrhart理论与多项式方法相结合，证明了D. Armstrong猜想同时存在的（a，b）-核中的盒子数是（a-1）（b-1）（a+b+1）/24。我们将P. Johnson的方法扩展到计算方差为ab（a-1）（b-1）（a+b）（a+b+1）/1440。通过将“同时核”和“盒数”的定义推广到仿射外尔群，我们将上述三个公式统一推广到单缀仿射型上.我们用H的“奇怪公式”进一步解释了数字24的出现。Freudenthal和H.德·弗里斯
英文摘要：
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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