admit representations in any base $p$ (assuming $p>1$). Similarly, given a partition $\lambda$, we will iterate the Littlewood decomposition at $p$. We define recursively $\lambda^{(i_1,\cdots,i_d)} = \lambda^{(i_1,\cdots,i_{d-1})(i_{d})}$. For each tuple in $\{0,1,\cdots,p-1\}^d$, this defines a partition. Note that by convention $\lambda^{()} = \lambda$. We also define $\lambda^{[i_1,\cdots,i_d]} = \lambda^{(i_1,\cdots,i_d)[]}$ (*i.e.* the core of that particular quotient).

We see these two definitions as two mappings, from $S= \{\{0,\cdots,p-1\}^d:d \in \mathbb{N}\}$ to either partitions or $p$-cores. We see elements of $S$ as indices for the vertices of the $p$-ary infinite rooted tree[^5]: recursively, the index $(i_1,\cdots,i_d)$ corresponds to the $i_d^\text{th}$ child of the vertex indexed by $(i_1,\cdots,i_{d-1})$, with the index $()$ mapped to the root. This means $\lambda^{(i_1,\cdots,i_d)}$ (resp. $\lambda^{[i_1,\cdots,i_d]}$) provides a labelling of the vertices of the $p$-ary infinite rooted tree by partitions (resp. $p$-cores), called the *$p$-quotient tower of $\lambda$* (resp. *$p$-core tower of $\lambda$*). We can deduce from that all but finitely many labels of the $p$-core tower or the $p$-quotient tower will be trivial partitions. An example is presented in Figure \[fig.tree\].

[Cp[2cm]{}C]{} && [$\phi:=\substack{\Tree [.{{{$\substack{()\\\cdots00{{\textbf{|}}}11\cdots\\()}$}}} {{$\phi$}} {{$\phi$}} {{$\phi$}} ]}$]{}

We can now state the following lemma, where $v_p(n)$ for the largest exponent $\alpha$ such that $p^\alpha$ divides 