Euler's partition theorem and the combinatorics of -sequences

Can revise bijection
To insert a k + a k+1 into (λ 1 , λ 2 , λ 3 , λ 4 , . . .) either do
(i) (λ 1 + a k , λ 2 + a k−1 , λ 3 , λ 4 , . . .)
or
(ii) (λ 1 + (a k − a k−1 ), λ 2 + (a k−1 − a k−2 ),
How to decide?
insert (a k−1 + a k−2 )into(λ 3 , λ 4 , . . .))
Do (i) if it does not create a carry in Fraenkel arithmetic;
otherwise do (ii).
So, insertion becomes a 2-d form of Fraenkel arithmetic.