Let a be an integer, and let p(x1, x2, ... , xn) = ∑1n k xk. Show that the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a, with all xi > 0 equals the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a - n(n + 1)/2, with all xi ≥ 0.
Solution
If xi is a positive integral solution to p = a, then xi-1 is a non-negative integral solution to p = a - n(n+1)/2. Conversely if xi is a non-negative integral solution to p = a - n(n+1)/2, then xi+1 is a positive integral solution to p = a. So we have a bijection between the two solution sets.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03