Problem 281

Problem 281

Pizza Toppings

You are given a pizza (perfect circle) that has been cut into m·n equal pieces and you want to have exactly one topping on each slice.

Let f(m,n) denote the number of ways you can have toppings on the pizza with m different toppings (m ≥ 2), using each topping on exactly n slices (n ≥ 1).
Reflections are considered distinct, rotations are not.

Thus, for instance, f(2,1) = 1, f(2,2) = f(3,1) = 2 and f(3,2) = 16. f(3,2) is shown below:

Find the sum of all f(m,n) such that f(m,n) ≤ 1015.



如果你有m种不同的馅料(m ≥ 2),每种馅料恰好可以放在n片披萨上(n ≥ 1),记f(m,n)为所有不同的馅料分配方案。

因此,举例来说,f(2,1) = 1,f(2,2) = f(3,1) = 2,以及f(3,2) = 16。f(3,2)的所有方案如下所示:

求所有满足f(m,n) ≤ 1015的f(m,n)之和。