  We study a particular class of infinite-dimensional representations of
$\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a
positive integer $p$, and are the lowest component in the $p$-fold tensor
product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We
construct a new polynomial basis for $L_n(p)$ arising from the embedding
$\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of
$L_n(p)$ are labelled by semi-standard Young tableaux, and are expressed as
Clifford algebra valued polynomials with integer coefficients in $np$
variables. Using combinatorial properties of these tableau vectors it is
deduced that they form indeed a basis. The computation of matrix elements of a
set of generators of $\mathfrak{osp}(1|2n)$ on these basis vectors requires
further combinatorics, such as the action of a Young subgroup on the horizontal
strips of the tableau.
