8 Combinatorics of Permutations, Second Edition
The obvious question that probably crossed the mind of the reader by now
is whether there exists an explicit formula for the numbers A(n, k). The
answer to that question is in the affirmative, though the formula contains a
summation sign. This formula is more difficult to prove than the previous
formulae in this Section.
THEOREM 1.11
For all nonnegative integers n and k satisfying k ≤ n, we have
A(n, k)=
k
i=0
(−1)
i
n +1
i
(k − i)
n
. (1.3)
While this theorem is a classic (it is more than a hundred years old), we
could not find an immaculately direct proof for it in the literature. Proofs
we did find used generating functions, or manipulations of double sums of
binomial coefficients, or inversion formulae to obtain