Schedule for: 16w5048 - Workshop in Analytic and Probabilistic Combinatorics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

By means of an asymptotic analysis of generating functions, we determine the limiting distribution of the order of the automorphism group of a random labeled tree. To be precise, we show that the logarithm of the number of automorphisms, suitably renormalized, converges weakly to a standard normal distribution. This result is also further extended to other random tree models.

We present a new proof of a result from Erdős and Rényi on the enumeration of graphs where a given subgraph is forbidden. It is based on generating function manipulations. Joint work in progress, with Gwendal Collet, Danièle Gardy, Bernhard Gittenberger and Vlady Ravelomanana.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lattice paths are very classic objects in both probability theory and enumerative combinatorics. In particular, weighted models bridge the gap between the two approaches very neatly. We consider an example, the Gouyou-Beauchamps model of lattice walks in the first quadrant, and discuss how to determine asymptotic enumeration formulas parameterized by the weights. The major tool is the theory of analytic combinatorics in several variables (ACSV) and we identify six different kinds of asymptotic regimes (called universality classes) which arise according to the values of the weights. Because we are able to explicitly and generically compute the constants of the asymptotic formula, we can determine a formula for a family of discrete harmonic functions. Furthermore, we are able to demonstrate an infinite class of models for which the counting generating function is not D-finite.

We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We compare our bounds with the existing bounds on the numbers of linear extensions. Joint work with Igor Pak and Greta Panova.

A P\'olya urn is a stochastic process that describes the composition of an urn that contains balls of different colours. The set of colours is usually a finite set $\{1, \ldots, d\}$. At each discrete-time step, one draws a ball uniformly at random in the urn (let $c$ be its colour), and replace it in the urn together with R_{c,i} balls of colour $i$, for all $1\leq i\leq d$ In this talk, I will present a generalisation of this model to an infinite, and potentially uncountable set of colours. In this new framework, the composition of the urn is a measure (possibly non-atomic) on a Polish space.

Let $[n] = \{1, \dots, n\}$ and let $\Omega_n$ be the set of all mappings from $[n]$ to itself. Let $f$ be a random uniform element of $\Omega_n$ and let $\mathbf{T}(f)$ and $\mathbf{B}(f)$ denote, respectively, the least common multiple and the product of the length of the cycles of $f$. Harris proved in 1973 that $\log \mathbf{T}$ converges in distribution to a standard normal distribution and, in 2011, E. Schmutz obtained an asymptotic estimate on the logarithm of the expectation of $\mathbf{T}$ and $\mathbf{B}$ over all mappings on $n$ nodes. We obtain analogous results for random uniform mappings on $n = kr$ nodes with preimage sizes restricted to a set of the form $\{0,k\}$, where $k = k(r) \geq 2$. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form $x^k + a$ over the integers modulo $p$, with $ p \equiv 1 \pmod k$. We also exhibit and discuss our numerical results on this heuristic. Joint work with R. Martins, C. Qureshi and E. Schmutz.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.