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?