# 2-quotient of integer partition

This question is mostly about understanding the notation used in the following article: Alex Eskin, Andrei Okounkov, Pillowcases and quasimodular forms, in: Victor Ginzburg (ed.), Algebraic Geometry and Number Theory in Honor of Vladimir Drinfeld's 50th Birthday, Birkhäuser 2006.

page 8 about the 2-quotients of a partition. It claims that every partition $${\bf{\lambda}}:=(\lambda_1, \lambda_2,\ldots,)$$ has two partitions $$\bf{\alpha, \beta}$$ such that

$$\Big\{\lambda_i -1 +1/2\Big\}=\Big\{\alpha_i -i+1/2 +\bar{p}_0(\lambda) \Big\}\sqcup \Big\{\beta_i -i+1/2+\bar{p}_0(\lambda) \Big\}$$

I mostly interested in balanced partition so I can forget about $$\bar{p}_0(\lambda)$$. I don't understand what is the definition of $$\alpha_i, \beta_i$$. Is it related to Frobenius coordinates?

I suspect $$\alpha_i$$ and $$\beta_i$$ refer to heights in the Russian way to describe Young diagrams (rotate the English notation 135 degrees), this makes it into a piecewise linear function, and this interpretation has a few nice applications.