Euler's partition theorem and the combinatorics of -sequences

l ≥ 2
The l-Euler theorem [BME2]: The number of partitions of an
integer N into parts from the set
{a (l)
0
+ a (l)
1 , a(l) 1
+ a (l)
2 , a(l) 2
+ a (l)
3 , . . .}
is the same as the number of partitions of N in which the ratio of
consecutive parts is greater than
Proof: via lecture hall partitions.
c l = l + √ l 2 − 4
2