Asymptotics of the Euler number of bipartite graphs

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.Comment: 13 pages, 6 figure, submitted to JCT

Descent pattern avoidance

We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern we show how to find the equation determining the spectrum. We give two length $4$ applications. First, we find the asymptotics of the number of permutations with no triple ascents and no triple descents. Second, we give the asymptotics of the number of permutations with no isolated ascents or descents. Our next result is a weighted pattern of length $3$ where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.Comment: 16 page

Parking cars of different sizes

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.Comment: 5 pages, 5 figue

Number of cycles in the graph of 312-avoiding permutations

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation $\pi = \pi_{1} \pi_{2} ... \pi_{n+1}$ there is a directed edge from the standardization of $\pi_{1} \pi_{2} ... \pi_{n}$ to the standardization of $\pi_{2} \pi_{3} ... \pi_{n+1}$. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.Comment: To appear in the Journal of Combinatorial Theory - Series