Euler's partition theorem and the combinatorics of -sequences

Theorem [Corteel,S 2004] Given positive integers s 1 , . . . , s n , the
generating function for the sequences λ 1 , . . . , λ n satisfying
is
λ 1
s 1
≥ λ 2
s 2
≥ · · · ≥ λ n−1
s n−1
≥ λ n
s n
≥ 0
∑ s2 −1 ∑ s3 −1
z 2 =0 z 3 =0 · · · ∑s n−1
s 1 z 2
z n=0 q⌈ s 2
∏ n
i=1 (1 − qb i )
⌉+ P n
i=2 z i
∏ n−1
i=2 qb i ⌈ z i+1 − z i ⌉
s i+1 s i
where b 1 = 1 and for 2 ≤ i ≤ n, b i = s 1 + s 2 + · · · + s i .
Corollary As q → 1, (1 − q) n × this gf −→
s 2 s 3 · · · s
∏ n
n
i=1 b .
i