Counting and Enumerating Crossing-free Geometric Graphs

We describe a framework for counting and enumerating various types of crossing-free geometric graphs on a planar point set. The framework generalizes ideas of Alvarez and Seidel, who used them to count triangulations in time $O(2^nn^2)$ where $n$ is the number of points. The main idea is to reduce the problem of counting geometric graphs to counting source-sink paths in a directed acyclic graph. The following new results will emerge. The number of all crossing-free geometric graphs can be computed in time $O(c^nn^4)$ for some $c < 2.83929$. The number of crossing-free convex partitions can be computed in time $O(2^nn^4)$. The number of crossing-free perfect matchings can be computed in time $O(2^nn^4)$. The number of convex subdivisions can be computed in time $O(2^nn^4)$. The number of crossing-free spanning trees can be computed in time $O(c^nn^4)$ for some $c < 7.04313$. The number of crossing-free spanning cycles can be computed in time $O(c^nn^4)$ for some $c < 5.61804$. With the same bounds on the running time we can construct data structures which allow fast enumeration of the respective classes. For example, after $O(2^nn^4)$ time of preprocessing we can enumerate the set of all crossing-free perfect matchings using polynomial time per enumerated object. For crossing-free perfect matchings and convex partitions we further obtain enumeration algorithms where the time delay for each (in particular, the first) output is bounded by a polynomial in $n$. All described algorithms are comparatively simple, both in terms of their analysis and implementation

Asymptotic Distributions for Block Statistics on Non-crossing Partitions

The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions. We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8. We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution

Stabilized-Interval-Free Permutations and Chord-Connected Permutations

Abstract. A stabilized-interval-free (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if sn denotes the number of SIF permutations on [n], S(z) = 1 + ∑ n≥1 snzn, and F (z) = 1 + ∑ n≥1 n!zn, then F (z) = S(zF (z)). This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the chordconnected permutations on [n], counted by {cn}n≥1, whose generating function satisfies S(z) = C(zS(z)). The expressions at hand have immediate probabilistic interpretations, via the celebrated moment-cumulant formula of Speicher, in the context of the free probability theory of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian. Résumé. Tel que défini par Callan, une permutation sur n chiffres est dite à intervalle stabilisé si elle ne stabilise pas d’intervalle propre de [n]. Ces permutations jouent le rôle des irréductibles dans la décomposition des permutations selon les partitions non-croisées. En d’autres mots, si sn dénombre les permutations à intervalle stabilisé sur

General position of a projection and its image under a free unitary Brownian motion

Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that $Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of this paper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say $\mu_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of $\mu_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \infty)$ distributions. This fact allows to compute the weight that $\mu_t$ assigns to $z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $SY_tSY_t^{\star}$.Comment: The letter `a' was used to denote two different objects: an operator and a real number. The operator is now denoted `S' referring to `symmetry

On the classification of easy quantum groups

In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have a deep connection to combinatorics. We continue their work on classifying these objects introducing some new examples of easy quantum groups. In particular, we show that the six easy groups O_n, S_n, H_n, B_n, S_n' and B_n' split into seven cases on the side of free easy quantum groups. Also, we give a complete classification in the half-liberated case.Comment: 39 pages; appeared in Advances in Mathematics, Vol. 245, pages 500-533, 201

Asymptotics of characters of symmetric groups, genus expansion and free probability

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S_q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q^{-1/2} converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.Comment: version 2: change of title; the section on Gaussian fluctuations was moved to a subsequent paper [Piotr Sniady: "Gaussian fluctuations of characters of symmetric groups and of Young diagrams" math.CO/0501112