The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\mathrm{\ell },1^n)$

Let $\mathfrak{o}$ be the valuation ring of a non-Archimedean local field with finite residue field. We give a procedure to find the representation zeta polynomial of ${\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathfrak{o}}({\mathfrak{o}}_{\mathrm{\ell }}\oplus {\mathfrak{o}}_1^{\oplus n})$ by induction on $n$. In particular, we show that the dimensions of the representations are given by evaluating finitely many polynomials at $q=\mathrm{\mid }{\mathfrak{o}}_1\mathrm{\mid }$.