For simplicity, consider an infinite locally-finite poset $\mathcal{P}$ with a unique bottom element $\perp$ whose finite order ideals obey a hook-length formula --- i.e. the number of linear extensions for each order ideal satisfy a (common) hook length formula. The examples I have in mind are

(1) the infinite square lattice: whose finite order ideals are exactly Young diagrams (the linear extensions of which are exactly Young Tableaux)

(2) the Young-Fibonacci lattice

(3) any infinite rooted tree: whose finite order ideals are among the finite rooted trees (the linear extensions of which are exactly the increasing trees)

Take a large but finite order ideal $I$ in $\mathcal{P}$ together with a large integer cut-off $0 << \, n \, << |I|$. Select with uniform probability a linear extension $l$ of $I$ and consider the order ideal $\text{res}_n(l)$ obtained by restricting it to the interval of values $[1 \dots n]$, namely

\begin{equation} \text{res}_n(l) \, := \, \Big\{x \in I \, \Big| \, l(x) \in [1 \dots n] \Big\} \end{equation}

This restriction map induces a probability measure $\mu_n$ on the space $\mathcal{I}_n$ of all order ideals in $\mathcal{P}$ of size $n$.

Imagine now that $l$ is a fixed linear extension. Let $\sigma: \mathcal{L}_I \longrightarrow \mathcal{L}_I$ denote the Schützenberger promotion operator and form the promotion-orbit

\begin{equation} \mathcal{O}_l \, := \, \Big\{\sigma^k \cdot l \, \Big| \, k \in \Bbb{Z} \Big\} \end{equation}

of $l$ and consider the restrictions $\text{res}_n \big( \sigma^k \cdot l \big)$ as $k$ varies; in this way we obtain another distribution $\rho_{n,l}$ on $\mathcal{I}_n$.

**Question:** What is the relationship between the distributions $\mu_n$ and $\rho_{n,l}$ as $|I| \rightarrow \infty$ in
view of Propp's concept of homomesy (as manifest by promotion) ?

regards,

A. Leverkühn