Euler's partition theorem and the combinatorics of -sequences

Theorem [Fraenkel 1985] Every nonnegative integer has a unique
(up to leading zeroes) representation as an l-ary string which does
not contain the pattern
(l − 1) (l − 2) ∗ (l − 1).
Proof Show that
is a bijection
f : b n b n−1 . . . b 1
−→
n∑
b i a i
i=1
f : T (l)
n
−→ {0, 1, 2, . . . a (l)
n+1 − 1}.
The l-representation of an integer x is
[x] = f −1 (x).