Svante Linusson

COMPLEXES OF NOT i-CONNECTED GRAPHS

Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by Vassiliev [38, 39, 41]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n−2)! spheres of dimension 2n−5. This answers a question raised by Vassiliev in connection with knot invariants. For this case the S_n-action on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n−2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n−3)-connected graphs we provide a formula for the generating function of the Euler characteristic.

A simple bijection for the regions of the Shi arrangement of hyperplanes

Abstract The Shi arrangement Ln is the arrangement of affine hyperplanes in R n of the form xi − xj = 0 or 1, for 1 ⩽ i R n into (n + 1)n−1 regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Ln containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements. It also implies the fact that the number of regions of Ln which are relatively bounded is (n − 1)n−1.

n! matchings, n! posets

We show that there are matchings on points without so-called left (neighbor) nestings. We also define a set of naturally labeled -free posets and show that there are such posets on elements. Our work was inspired by Bousquet-Melou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884-909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled -free posets, permutations avoiding a specific pattern, and so-called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Melou et al., and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled -free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections factors through certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53].

An inhomogeneous multispecies TASEP on a ring

We reinterpret and generalize conjectures of Lam and Williams as statements about the stationary distribution of a multispecies exclusion process on the ring. The central objects in our study are the multiline queues of Ferrari and Martin. We make some progress on some of the conjectures in different directions. First, we prove their conjectures in two special cases by generalizing the rates of the Ferrari-Martin transitions. Secondly, we define a new process on multiline queues, which have a certain minimality property. This gives another proof for one of the special cases; namely arbitrary jump rates for three species.

Doubly alternating Baxter permutations are Catalan

The Baxter permutations who are alternating and whose inverse is also alternating are shown to be enumerated by the Catalan numbers. A bijection to complete binary trees is also given.

On percolation and the bunkbed conjecture

We study a problem on edge percolation on product graphs G × K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. Every edge in G × K2 is present with probability p independent of other edges. The bunkbed conjecture states that for all G and p, the probability that (u, 0) is in the same component as (v, 0) is greater than or equal to the probability that (u, 0) is in the same component as (v, 1) for every pair of vertices u, v ∈ G. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs G, in particular outerplanar graphs.

Partitions with Restricted Block Sizes, Möbius Functions, and the k -of-Each Problem

Given a list of n real numbers, one wants to decide whether every number in the list occurs at least k times. It will be shown that

Thomassen's Choosability Argument Revisited

\Omega(n\log n)

Complexes of t -Colorable Graphs

is a sharp lower bound for the depth of an algebraic decision or computation tree solving this problem for a fixed k. For linear decision trees, the coefficient can be taken to be arbitrarily close to 1 (using the ternary logarithm). This is done by using the Bjorner--Lovasz--Yao method, which turns the problem into one of estimating the Mobius function for a certain partition lattice. The method will work also for the more general T-multiplicity problem when T is additive and cofinite. A formula for the exponential generating function for the Mobius function of a partition poset with restricted block sizes in general will also be given.

A counter-intuitive correlation in a random tournament

Thomassen (J. Combin. Theory Ser. B, 62 (1994), pp. 180-181) proved that every planar graph is 5-choosable. This result was generalized by Skrekovski (Discrete Math., 190 (1998), pp. 223-226) and He, Miao, and Shen (Discrete Math., 308 (2008), pp. 4024-4026), who proved that every