Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then, $$\sum_{\lambda\vdash n}f_{\lambda}^2=n! \tag1$$ where the sum runs through all partitions of $n$.
Given a restricted (or distinct parts) integer partition $\lambda$ of $n$, let $g_{\lambda}$ denote the number of standard Young tableaux of shape the shifted Young diagram $S(\lambda)$ of $\lambda$. Note that distinct parts also means parts that differ by $1$ or more. Then, $$\sum_{\lambda\vdash n}2^{n-\ell(\lambda)}g_{\lambda}^2=n! \tag2$$ where the sum runs through all partitions into distinct parts of $n$.
These two results allow for probability (Plancherel) measures in their respective environments.
QUESTION. Is there a result analogous to (1) and (2) for integer partitions whose parts differ by $2$ or more?
