Is there an intuitive way to define what the p-quotient of an integer partition is?


In response to the comments:
my vague understanding is that the p-quotient is somehow division with remainder generalized to partitions. They turn up in the classic by Littlewood that I was reading:
<http://rspa.royalsocietypublishing.org/content/209/1098/333.abstract>
See also Macdonalds book on symmetric functions and Hall polynomials.



p-quotient of an integer partition
I learned the following perspective in part from Bill Doran.


Redefine a partition through its boundary, extended along the axes in both directions: 


A *partition* is a bi-infinite sequence of vertical $(0,1)$ and horizontal $(1,0)$ steps $(... S\_{-2},S\_{-1},S\_0,S\_1,S\_2,...)$ whose prefix is all vertical steps, and whose suffix is all horizontal steps. A *square* of the partition is a pair of steps so that the first step is horizontal and the second step is vertical. The *hook length* of that square is the distance between the steps. 


Then the $p$-*quotient* of a partition is the $p$-tuple of partitions you get by the steps in each congruence class mod $p$: $P\_i = (...S\_{-2p+i},S\_{-p+i},S\_i,S\_{p+i},S\_{2p+i},...)$ for $i \in \{0,..., p-1\}$.


Each square of a partition of the $p$-quotient corresponds to a square of the original partition whose hook length is divisible by $p$. 


$p$-*cores* are partitions whose $p$-quotient is a $p$-tuple of empty partitions. They can be parametrized by an equivalence class of $p$-tuples of values $k\_i$ so that $S\_{p k + i}$ is vertical iff $k \le k\_i$, i.e., the offsets of the empty partitions.


If you strip $p$-rimhook after $p$-rimhook off of a partition, this always results in the same $p$-core, and the choices don't matter. The choices correspond to taking one corner square (hook length 1) at a time off of a partition in the $p$-quotient.


Although in some applications, $p$ is prime, that is not necessary. 


