be seen from Figure \[fig.hooks\].

In terms of the 01-sequence $(x_i)$, a hook of length $p$ in $\lambda$ corresponds to a pair $x_{i_0} = 1$, $x_{i_0+p} = 0$. Swapping these two entries in the sequence $(x_i)$ gives the 01-sequence of a partition of size decreased by $p$, due to the removal of a hook of length $p$. Starting with one partition $\lambda$, we can swap pairs at distance $p$ and iterate this procedure until no more $p$-hook can be found. The partition associated to this final 01-sequence is actually independent from the ordering of the hook removals (or the ordering of swaps of a $\cdots1\cdots 0\cdots$ into a $\cdots0\cdots 1\cdots$), and called the *$p$-core of $\lambda$*, which we denote $\lambda^{[]}$. We see in Figure \[fig.01\] that the 3-core of the partition $(18,7,6)$ is $(3,1)$.

We now define the $p$-quotients of the partition $\lambda$, for $p$ still not a prime. We start with $ (x_i) := x(\lambda) $, and consider the $p$-tuple of subsequences given by $((x_{p i +j} ): 0\le j \le p-1)$. For each $0\le j\le p-1$, this defines a 01-sequence and thus an associated partition, which we call the $j^{\text{th}}$ *$p$-quotient* of $\lambda$ and denote $\lambda^{(j)}$. We obtain 