
222 Combinatorics of Permutations, Second Edition
p
be the result of adding k to every element of that permutation. Then, by
construction, p = p
p
∈ S
n
and c(p)=ks + t = c as desired.
One can modify the proof of the previous theorem to locate precisely where
the internal zeros could be for an IZ sequence. We will need the fact (estab-
lished by computer) that for n ≤ 12 the only IZ integers are 6, 8, and 9, and
that they all satisfied the following result.
THEOREM 5.43
For any positive integer n, the sequence F
n
does not have internal zeros, except
possibly for c = M
n
− 1 or c = M
n
− 2, but not both.
PROOF We prove this theorem by induction on n. Numerical ...