Quiz # 10 for Dr. Z.’s Number Theory Course for Dec. 5, 2013

NAME: (print!) E-MAIL ADDRESS: (print!)

1. An extremely distinct partition of n is a sequence of integers (λ_{1}, λ_{2}, …, λ_{t}) ,

such that

λ_{1} + λ_{2} + … + λ_{t} = n ,

and

λ_{1} − λ_{2} ≥ 2 λ_{2} − λ_{3} ≥ 2 , …, , λ_{t−1} − λ_{t} ≥ 2

and

λ_{t} > 0 ,

Let q(n) be the number of partitions of n, and q(n, k) be the number of exteremely distinct partitions of n whose largest part is k.

(i) (5 points) Explain why

q(n, k) = ∑_{r=1}^{k−2} q(n − k, r) ,

and, of course

q(n, n) = 1 .

(ii) Use the above recurrence, and

q(n) = ∑_{k=1}^{n} q(n, k)

to compute q(n) for 1 ≤ n ≤ 5.