Pattern Avoidance for Random Permutations

Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences and a corresponding joint distribution of independent Bernoulli random variables, which as a corollary yields a Poisson approximation for the distribution of the number of occurrences of any pattern. We also investigate occurrences of consecutive patterns in random Mallows permutations, of which uniform random permutations are a special case. These bounds allow us to estimate the probability that a pattern occurs any number of times and, in particular, the probability that a random permutation avoids a given pattern.Comment: 24 pages, 2 Figures, 4 Table

Asymptotic distribution of fixed points of pattern-avoiding involutions

For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as well as all patterns of length 3. For monotone patterns we utilize the connection with standard Young tableaux with at most $k$ rows and involutions avoiding a monotone pattern of length $k$. For every pattern of length 3 we give the bivariate generating function with respect to fixed points for the involutions that avoid that pattern, and where applicable apply tools from analytic combinatorics to extract information about the limiting distribution from the generating function. Many well-known distributions appear.Comment: 16 page

Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph $\mathcal{P}(n,q)$ which results from taking the union of two paths of length $n\geq 1$, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter $0<q(n)\leq 1$. This random graph model, the tangled path, goes through an evolution: if $q$ is close to $0$ the graph bears resemblance to a path and as $q$ tends to $1$ it becomes an expander. In an effort to understand the evolution of $\mathcal{P}(n,q)$ we determine the treewidth and cutwidth of $\mathcal{P}(n,q)$ up to log factors for all $q$. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of $q$.Comment: 40 pages, 7 figure

Neural Connectivity with Hidden Gaussian Graphical State-Model

The noninvasive procedures for neural connectivity are under questioning. Theoretical models sustain that the electromagnetic field registered at external sensors is elicited by currents at neural space. Nevertheless, what we observe at the sensor space is a superposition of projected fields, from the whole gray-matter. This is the reason for a major pitfall of noninvasive Electrophysiology methods: distorted reconstruction of neural activity and its connectivity or leakage. It has been proven that current methods produce incorrect connectomes. Somewhat related to the incorrect connectivity modelling, they disregard either Systems Theory and Bayesian Information Theory. We introduce a new formalism that attains for it, Hidden Gaussian Graphical State-Model (HIGGS). A neural Gaussian Graphical Model (GGM) hidden by the observation equation of Magneto-encephalographic (MEEG) signals. HIGGS is equivalent to a frequency domain Linear State Space Model (LSSM) but with sparse connectivity prior. The mathematical contribution here is the theory for high-dimensional and frequency-domain HIGGS solvers. We demonstrate that HIGGS can attenuate the leakage effect in the most critical case: the distortion EEG signal due to head volume conduction heterogeneities. Its application in EEG is illustrated with retrieved connectivity patterns from human Steady State Visual Evoked Potentials (SSVEP). We provide for the first time confirmatory evidence for noninvasive procedures of neural connectivity: concurrent EEG and Electrocorticography (ECoG) recordings on monkey. Open source packages are freely available online, to reproduce the results presented in this paper and to analyze external MEEG databases

A central limit theorem for descents of a Mallows permutation and its inverse

This paper studies the asymptotic distribution of descents \des(w) in a permutation $w$, and its inverse, distributed according to the Mallows measure. The Mallows measure is a non-uniform probability measure on permutations introduced to study ranked data. Under this measure, permutations are weighted according to the number of inversions they contain, with the weighting controlled by a parameter $q$. The main results are a Berry-Esseen theorem for \des(w)+\des(w^{-1}) as well as a joint central limit theorem for (\des(w),\des(w^{-1})) to a bivariate normal with a non-trivial correlation depending on $q$. The proof uses Stein's method with size-bias coupling along with a regenerative process associated to the Mallows measure.Comment: v2 some added references and minor changes to introduction. 35 pages, 1 figure, 1 table. Comments are welcome

Cycles in Mallows random permutations

We study cycle counts in permutations of (Formula presented.) drawn at random according to the Mallows distribution. Under this distribution, each permutation (Formula presented.) is selected with probability proportional to (Formula presented.), where (Formula presented.) is a parameter and (Formula presented.) denotes the number of inversions of (Formula presented.). For (Formula presented.) fixed, we study the vector (Formula presented.) where (Formula presented.) denotes the number of cycles of length (Formula presented.) in (Formula presented.) and (Formula presented.) is sampled according to the Mallows distribution. When (Formula presented.) the Mallows distribution simply samples a permutation of (Formula presented.) uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means (Formula presented.). Here we show that if (Formula presented.) is fixed and (Formula presented.) then there are positive constants (Formula presented.) such that each (Formula presented.) has mean (Formula presented.) and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when (Formula presented.) there is a striking difference between the behavior of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behavior depends on the parity of (Formula presented.) when (Formula presented.). Both (Formula presented.) and (Formula presented.) have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all (Formula presented.). We describe these limiting distributions in terms of Gnedin and Olshanski's bi-infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as (Formula presented.) the expected number of 1-cycles tends to (Formula presented.) —which, curiously, differs from the value corresponding to (Formula presented.). In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for (Formula presented.) and (Formula presented.) odd versus (Formula presented.) even.</p