Polynomial expressions for the dimensions of the representations of symmetric groups and restricted standard Young tableaux

Given a partition $\lambda$ of a number $k$, it is known that by adding a long line of length $n-k$, the dimension of the associated representation of $S_n$ is an integer-valued polynomial of degree $k$ in $n$. We show that its expansion in the binomial basis is bounded by the length of $\lambda$, and that the resulting coefficient of index $h$, with alternating signs, counts the standard Young tableaux of shape $\lambda$ in which a given collection of consecutive $h$ numbers lie in increasing rows. We also construct bijections in order to demonstrate explicitly that this number is indeed independent of the set of consecutive $h$ numbers used.