Let $\lambda_0,\lambda_1,\lambda_2,\lambda_3,\ldots$ be a sequence of Young diagrams, such that each successive diagram is obtained from the prior by the addition of one box (don't forget that the row lengths must always be weakly decreasing). For example such a sequence could arise from a Plancherel growth process.

Denote by $f_\lambda$ the number of standard Young tableau of shape $\lambda$. I'm wondering if there's a representation-theoretic interpretation of the sum:

$$f(x)=\sum_{k\geq 0}f_{\lambda_{k}}x^k.$$

Motivation: I'm encountering a sum of determinants which seems to have the above structure and I'm wondering if there's a general method or theory behind the evaluation of above functions $f(x)$. I've never seen such a sum before where the "dimensions of each representation" are increasing.