  Fix an integer $d\geq 1$. Solomon counts index-$n$ submodules of the free
module $R^d$ for any Dedekind domain $R$ that admits a Dedekind zeta function.
The proof involves the Hermite normal form for matrices. We propose another
normal form in the case where $R=\mathbb{F}_q[[T]]$, which unlike the Hermite
normal form, recovers the Smith normal form. We use our normal form to give a
uniform approach to Solomon's result and a refinement of it due to
Petrogradsky, which counts finite-index submodules $M$ of $R^d$ such that
$R^d/M$ has a given module structure. Our normal form and Hermite's can be
interpreted as reduced Gr\"obner bases with respect to two different monomial
orders. Counting with our normal form gives rise to a ``total distance''
statistics $W$ and certain ``spiral shifting'' operators $g_1,\dots,g_d$ on the
set $X=\mathbb{N}^d$ of $d$-tuples of nonnegative integers. These combinatorial
structures are of independent interest. We show that these operators commute,
and collectively act on $X$ freely and transitively, which in particular gives
a nontrivial self-bijection on $\mathbb{N}^d$. We give a formula on how $g_j$
interacts with the $W$-statistics on $X$, leading to a rational formula for the
generating function for $W$-statistics over any forward orbit in $X$ under
these operators.
