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](https://books.google.com.au/books?id=gGUTVxVieRIC&pg=PA8&lpg=PA8&dq=2+quotient+of+a+balanced+partition&source=bl&ots=ZjzWEKbbd_&sig=ACfU3U0LWv7OxdVEjvwqc2qil4Qir3UrOw&hl=en&sa=X&ved=2ahUKEwjfnYOjg5HnAhWFWisKHfc6CasQ6AEwAnoECAgQAQ#v=onepage&q=2%20quotient%20of%20a%20balanced%20partition&f=false).


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? 



2-quotient of integer partition
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.


Information regarding quotients, (and its relation to character values in particular), can [be found here](https://www.math.upenn.edu/~peal/polynomials/borderStripTableaux.htm).
There is the article 


*Olivier Brunat and Rishi Nath. Cores and quotients of partitions through the Frobenius symbol. ArXiv e-prints, 2019.* 


which apparently does what the title suggest, in case the Frobenius description of partitions is relevant.


