A Cayley graph for F2×F2F_{2}\times F_{2} which is not minimally almost convex

We give an example of a Cayley graph $\Gamma$ for the group $F_{2}\times F_{2}$ which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for $F_{2}\times F_{2}$ does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property $K$ lying between FFTP and MAC (i.e., $\text{FFTP}\Rightarrow K\Rightarrow\text{MAC}$) is dependent on the generating set. This includes the well known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Po\'{e}naru's condition $P(2)$ and the basepoint loop shortening property for which dependence on the generating set was previously unknown. We also show that the Cayley graph $\Gamma$ does not have the loop shortening property, so this property also depends on the generating set.Comment: 11 pages, 2 figure

Counting Planar Eulerian Orientations

Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - \mu x)/\log(1 - \mu x),$ where we conjecture that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices.Comment: 26 pages, 20 figure

Off-critical parafermions and the winding angle distribution of the O(nn) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $\gamma_1$ and $\gamma_{11}$ as well as wedge critical exponents $\gamma_2(\alpha)$ and $\gamma_{21}(\alpha)$ and the exponent characterising the winding angle distribution of the O($n$) model in the half-plane, or more generally in a wedge of wedge-angle $\alpha.$ We assume only the existence of these exponents and, for some values of $n,$ the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality $\gamma_1-\gamma_{11} \ge 1,$ and its extension $\gamma_2(\alpha)-\gamma_{21}(\alpha) \ge 1$ for the edge exponents. We provide conjectured values for all exponents for $n \in [-2,2).$Comment: 17 pages, 5 figures, revised versio

Staircases, dominoes, and the growth rate of 1324-avoiders

We establish a lower bound of 10.271 for the growth rate of the permutations avoiding 1324, and an upper bound of 13.5. This is done by first finding the precise growth rate of a subclass whose enumeration is related to West-2-stack-sortable permutations, and then combining copies of this subclass in particular ways

The generating function of planar Eulerian orientations

37 pp.International audienceThe enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model.We solve both problems -- namely the enumeration of planarEulerian orientations and of 4-valent planar Eulerian orientations --by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, $\mu^n /(n \log n)^2$, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order 2. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model.Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices.In the 4-valent case, we also observe an unexpected connection with theenumeration of maps equipped with a spanning tree that is internallyinactive in the sense of Tutte. This connection remains to beexplained combinatorially