The Brownian limit of separable permutations

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion

Permutations avoiding 1324 and patterns in Łukasiewicz paths

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution

Sorting probability of Catalan posets

We show that the sorting probability of the Catalan poset} $P_n$ satisfies $\delta(P_n)= O \bigl(n^{-5/4}\bigr)$.Comment: 10 pages, 6 figure

Strongly Correlated Random Interacting Processes

The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including selfinteracting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations

Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

For $\eta\in S_3$, let $S_n^{\text{av}(\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\eta$, and let $E_n^{\text{av}(\eta)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}(\eta)}$. For $n\ge k\ge2$ and $\tau\in S_k^{\text{av}(\eta)}$, let $N_n^{(k)}(\sigma)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $\sigma\in S_n^{\text{av}(\eta)}$, and let $N_n^{(k;\tau)}(\sigma)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $\tau$. We obtain explicit formulas for $E_n^{\text{av}(\eta)}N_n^{(k;\tau)}$ and $E_n^{\text{av}(\eta)}N_n^{(k)}$, for all $2\le k\le n$, all $\eta\in S_3$ and all $\tau\in S_k^{\text{av}(\eta)}$. These exact formulas then yield asymptotic formulas as $n\to\infty$ with $k$ fixed, and as $n\to\infty$ with $k=k_n\to\infty$. We also obtain analogous results for $S_n^{\text{av}(\eta_1,\cdots,\eta_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns $\{\tau_i\}_{i=1}^r$, where $\tau_i\in S_{m_i}$, in the case that $\{\tau_i\}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, $\tau_1=2413,\tau_2=3142$.Comment: A number of typos have been correcte