Let $\Lambda_n$ denote the lattice of all integer partitions of size $n$ where the operative partial order $\lambda \unlhd \mu $ is the dominance order defined by
\begin{equation} \lambda_1 + \cdots + \lambda_k \ \leq \ \mu_1 + \cdots + \mu_k \end{equation}
for all $k \geq 1$. I'll view a partition $\lambda$ is viewed as semi-infinite, non-increasing, non-negative integer sequence $\lambda = \big(\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \cdots \big)$ whose size $|\lambda| = \lambda_1 + \lambda_2 + \lambda_3 + \cdots$ is finite and equal to $n$.
Question 1: Does anyone know a formula for the number of order ideals in $\Lambda_n$ ?
Question 2: By RSK, each partition $\lambda \in \Lambda_n$ corresponds bijectively to an involution in the symmetric group $S_n$. In light of RSK, do the order ideals of $\Lambda_n$ correspond bijectively to a class of (interesting) elements in $S_n$ ?
Comment: This second Question works on a misunderstanding and is the result of some hasty reasoning. Involutions in $S_n$ are not in bijection with partitions $\lambda \in \Lambda_n$ but rather in bijection with standard young tableaux of partitions $\lambda \in \Lambda_n$. One way of rectifying the question is to look at the poset $\mathcal{T}_n$ of all standard Young tableaux of all partitions $\lambda \in \Lambda_n$ under the (induced) dominance order and ask for an interpretation of its order ideals. Recall that if we view two standard Young Tableaux $T$ and $S$ of respective shapes $\lambda$ and $\mu$ as two saturated chains $\big( \lambda^{(0)} \subset \cdots \subset \lambda^{(n)} \big)$ and $\big(\mu^{(0)} \subset \cdots \subset \mu^{(n)} \big)$ in the Young Lattice with $\lambda^{(0)} = \mu^{(0)}= \emptyset$ as well as $\lambda^{(n)}= \lambda$ and $\mu^{(n)}=\mu$ then $T \unlhd S$ if and only if $\lambda^{(k)} \unlhd \mu^{(k)}$ for $0 \leq k \leq n$.
Apologies.
thanks, ines.