155 research outputs found
Asymptotics of the Euler number of bipartite graphs
We define the Euler number of a bipartite graph on vertices to be the
number of labelings of the vertices with 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 with the path 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 , we numerically solve the eigenvalue problem.Comment: 13 pages, 6 figure, submitted to JCT
Cyclotomic factors of the descent set polynomial
We introduce the notion of the descent set polynomial as an alternative way
of encoding the sizes of descent classes of permutations. Descent set
polynomials exhibit interesting factorization patterns. We explore the question
of when particular cyclotomic factors divide these polynomials. As an instance
we deduce that the proportion of odd entries in the descent set statistics in
the symmetric group S_n only depends on the number on 1's in the binary
expansion of n. We observe similar properties for the signed descent set
statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up
notatio
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
A combinatorial proof of the log-concavity of the numbers of permutations with runs
We combinatorially prove that the number of permutations of length
having runs is a log-concave sequence in , for all . We also give
a new combinatorial proof for the log-concavity of the Eulerian numbers.Comment: 10 pages, 4 figure
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