Random permutations from the staircase in the Okada monoid     permutations

The Okada monoid $\mathcal{O}_N$ is generated by $e_1,\dots,e_{N-1}$ subject to $e_i^2=e_i$, $e_ie_j=e_je_i$ for $|i-j|\ge 2$, and $e_{i+1}e_ie_{i+1}=e_{i+1}$; it is Okada's algebra at $x_i=y_i=1$. Hivert and Scott realize $\mathcal{O}_N$ by labelled non-crossing arc diagrams and show $|\mathcal{O}_N|=N!$, the elements being $e_\sigma=e_{i_1}\cdots e_{i_k}$ for the lexicographically minimal reduced word $\sigma=s_{i_1}\cdots s_{i_k}$, $\sigma\in S_N$.

Random element. Take the staircase diamond diagram with $N-1$ rows, whose reading word $(1)(2\,1)(3\,2\,1)\cdots(N{-}1\,\cdots\,1)$ is a reduced word of the longest permutation. Each box, in row $i$, is independently a double U-turn (the letter $e_i$, shaded) with probability $p$ and a double straight square (the letter is deleted) with probability $1-p$. The product of the surviving letters in reading order is $e_\sigma$ for a unique $\sigma\in S_N$, which is plotted as the points $(i,\sigma(i))$. This is the Okada analogue of the staircase pipe dreams of [MPPY], where the same staircase is evaluated in the 0-Hecke monoid.

Loop picture and arc diagram. The strands enter on the left at levels $1,\dots,N$ and leave on the right at levels $\bar 1,\dots,\bar N$, numbered from the bottom. Their connectivity is a non-crossing perfect matching of these $2N$ endpoints; each arc carries the lowest level its path reaches. Closed loops are discarded, since $e_i^2=e_i$. Hover over a strand or an arc to highlight it in both pictures.

Decoding. Let $D$ be the labelled arc diagram. If $N$ and $\bar N$ are joined by an arc of label $N$, put $d=N$ and delete that arc. Otherwise let $d$ be the largest index such that $\bar d$ and $\overline{d+1}$ are joined by an arc of label $d$; then $D=D^\flat e_{N-1}\cdots e_d$ (Hivert–Scott, Prop. right-code-factor). In both cases $N-d$ is the last entry of the Lehmer-type code of $\sigma$, and the procedure recurses on $D^\flat\in\mathcal{O}_{N-1}$.

For $N\le 10$ the staircase is drawn; for larger $N$ only the permutation is shown. The random numbers are reused when $p$ changes, so moving $p$ only flips boxes monotonically; Resample draws new ones. Nontrivial limit shapes appear when $1-p$ is of order $1/N$.