Euler's partition theorem and the combinatorics of -sequences

Θ (l)
n : Bijection for the l-Lecture Hall Theorem
Given a partition µ into parts in
{a 0 + a 1 , a 1 + a 2 , a 2 + a 3 , . . . , a n−1 + a n }
construct λ = (λ 1 , λ 2 , . . . , λ n ) by inserting the parts of µ in nonincreasing
order as follows:
To insert a k−1 + a k into (λ 1 , λ 2 , . . . , λ n ):
If k = 1, then add a 1 to λ 1 ;
otherwise, if (λ 1 + a k − a k−1 ) ≥ (a n /a n−1 )(λ 2 + a k−1 − a k−2 ),
add a k − a k−1 to λ 1 , add a k−1 − a k−2 to λ 2 ;
recursively insert a k−2 + a k−1 into (λ 3 , λ 4 , . . . , λ n )
otherwise,
add a k to λ 1 , and add a k−1 to λ 2 .