Use $\text{perm}_t$ to denote the set of all permutations among $t$ items. For any particular subset $S\subseteq\{0,1\}^n$ and any $\sigma\in \text{perm}_t$, we define \begin{align} P_S(\sigma) = \sum_{x_1,\ldots,x_t\in S}|{x_{\sigma^{-1}(1)},\ldots,x_{\sigma^{-1}(t)}}\rangle \langle{x_1,\ldots,x_t}|. \end{align}
Then, define
\begin{align} \Pi_{\text{sym}}^{S,t}=\frac{1}{t!}\sum_{\sigma~\in~\text{perm}_t}P_S(\sigma) \end{align} to be the projector onto the symmetric subspace of $(\mathbb{C}^{S})^{\otimes t}$.
I am trying to compute
\begin{equation} \underset{S}{\mathbb{E}}\left[\frac{\Pi_{\text{sym}}^{S,t}}{\mathsf{Tr}\big(\Pi_{\text{sym}}^{S,t}\big)}\right], \end{equation} where the expectation is taken over a random choice all subsets $S$ of a particular size $K$, for $t=1$, $t=2$, and $t=3$.
The case when $t=1$ is relatively simple. Note that for any choice of $S$, \begin{equation} \Pi_{\text{sym}}^{S,1} = P_S(\sigma) = \sum_{x\in S} ~|{x}\rangle\langle{x}|. \end{equation} So, for any choice of $S$, \begin{equation} \text{Tr}(\Pi_{\text{sym}}^{S,1}) = |S|. \end{equation} Now, fix $|S| = K$. \begin{equation} \begin{aligned} \underset{S}{\mathbb{E}}\left[\Pi_{\text{sym}}^{S,1}\right] &= \underset{S~\text{with size}~K}{\mathbb{E}}\left[\sum_{x\in S} ~|{x}\rangle\langle{x}|\right] \\ &= \frac{\binom{2^n - 1}{K-1}}{\binom{2^n}{K}} \mathbb{I}_{n} \\ &= \frac{K}{2^n} \cdot \mathbb{I}_n. \end{aligned} \end{equation} where $\mathbb{I}_{n}$ is the maximally mixed state on $n$ qubits. Hence, for subsets $S$ with $|S| = K$, \begin{equation} \begin{aligned} \underset{S}{\mathbb{E}}\left[ \frac{\Pi_{\text{sym}}^{S,1}}{\mathsf{Tr}(\Pi_{\text{sym}}^{S,1})}\right] &= \frac{\frac{K}{2^n} \cdot \mathbb{I}_n}{K} \\ &= \frac{ \mathbb{I}_n}{2^n}. \end{aligned} \end{equation}
However, cases for which $t=2$ and $t=3$ seem more complicated.
