### Background


Recall that the *$q$-analogue* $[n]\_q\in\mathbb Z[q]$ of a natural number $n\in\mathbb N$ is defined as
$$ [n]\_q := \frac{q^n -1}{q-1}$$
the idea being that formulas involving $q$ will specialize along $\mathbb Z[q]/(q-p^n)$ to counting formulas about finite fields, while specializing along $\mathbb Z[q]/(q-1)$ will yield counting formulas about finite sets.


For example, defining the $q$-factorial by
$$[n]\_q! := [1]\_q [2]\_q \dotsm [n]\_q,$$
we have that the number of $k$-dimensional subspaces of $\mathbb F\_q^n$ is
$$\#\mathrm{Gr}\_{n,k}(\mathbb F\_q) = \binom nk\_q :=\frac{[n]\_q!}{[n-k]\_q![k]\_q!}$$
and $\binom nk\_q$ reduces to $\binom nk$ when $q\to 1$.


### Question


Now let $p$ be a fixed prime. In my work, I've come across the expressions
$$ [p]\_q^{k\_1} [p^2]\_q^{k\_2} \dotsm [p^n]\_q^{k\_n} $$
with $k\_i\ge0$, $i=1,\dotsc, n$, as well as funny things like
$$ [p^r]\_{q^{p^s}} = \frac{q^{p^{r+s}}-1}{q^{p^s}-1}$$


**Do these have any known combinatorial interpretation?**


I will usually want to consider the above product when the values of $n$ and $\sum k\_i$ are fixed, so we can consider it ranging over all partitions of $\sum k\_i$ into $n$ non-negative integers.


### Motivation


Ultimately I'm interested in the specialization $\mathbb Z[q] \to \mathbb Z\_p[\![q-1]\!] \to \mathbf A\_{\mathrm{inf}}(R)$, where $R$ is a perfectoid ring containing a compatible choice of roots of unity $\zeta\_{p^\infty}$. Letting $$\epsilon = (1,\zeta\_p,\dots) \in R^\flat,$$ the structure map $\mathbb Z[\![q-1]\!] \to \mathbf A\_{\mathrm{inf}}$ is given by $q\mapsto[\epsilon]$. Then the Fontaine map $$\tilde\theta\_r\colon \mathbf A\_{\mathrm{inf}}(R) \to W\_r(R)$$ has kernel generated by $[p^r]\_q$ ([Example 3.16](https://arxiv.org/pdf/1602.03148.pdf)), while the map $\theta\_r = \tilde\theta\_r \varphi^r$ has kernel generated by $[p^r]\_{q^{1/p^r}}$. You can get the product of these ideals, above, out of some $RO(S^1)$-graded THH calculations.



Product of $q$-analogues
Recall Legendre's formula


$$ v\_p(n!) = \sum\_{s=1}^\infty\left\lfloor\frac n{p^s}\right\rfloor = \sum\_{r=0}^\infty a\_r[r]\_p $$
where $n = \sum a\_r p^r$ is the base-$p$ expansion of $n$.


A $q$-analogue of this formula is provided by Lemma 4.8 of [The $p$-completed cyclotomic trace in degree 2](https://arxiv.org/abs/1907.10530v1), which states that


\begin{equation}\label{AClB}\tag{$\ast$} \large [n]\_q! = u\prod\_{r=1}^\infty \varphi^{r-1}([p]\_q)^{\lfloor n/p^r\rfloor} \end{equation}


for a unit $u\in\mathbb Z\_p[\![q-1]\!]^\times$. While my computations yielded products
$$ [p]\_q^{k\_1} [p^2]\_q^{k\_2} \dotsm [p^n]\_q^{k\_n}, $$
for arbitrary $k\_i\ge0$, the ones most relevant to my eventual application (the regular slice filtration on THH) were the principal ideal generated by the RHS of \eqref{AClB} (when $p$ divides $n$).


