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

Given a partition λ 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 λ, and that the resulting coefficient of index h, with alternating signs, counts the standard Young tableaux of shape λ 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.