1,644 research outputs found
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
Introduction to Partially Ordered Patterns
We review selected known results on partially ordered patterns (POPs) that
include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified)
maxima and minima) in permutations, the Horse permutations and others. We
provide several (new) results on a class of POPs built on an arbitrary flat
poset, obtaining, as corollaries, the bivariate generating function for the
distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and
Pell numbers, as well as generalizations of few results in the literature
including the descent distribution. Moreover, we discuss q-analogue for a
result on non-overlapping segmented POPs. Finally, we suggest several open
problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea
Two problems on independent sets in graphs
Let denote the number of independent sets of size in a graph
. Levit and Mandrescu have conjectured that for all bipartite the
sequence (the {\em independent set sequence} of ) is
unimodal. We provide evidence for this conjecture by showing that is true for
almost all equibipartite graphs. Specifically, we consider the random
equibipartite graph , and show that for any fixed its
independent set sequence is almost surely unimodal, and moreover almost surely
log-concave except perhaps for a vanishingly small initial segment of the
sequence. We obtain similar results for .
We also consider the problem of estimating for
in various families. We give a sharp upper bound on the number of
independent sets in an -vertex graph with minimum degree , for all
fixed and sufficiently large . Specifically, we show that the
maximum is achieved uniquely by , the complete bipartite
graph with vertices in one partition class and in the
other.
We also present a weighted generalization: for all fixed and , as long as is large enough, if is a graph on
vertices with minimum degree then with equality if and only if
.Comment: 15 pages. Appeared in Discrete Mathematics in 201
Enumerating two permutation classes by number of cycles
We enumerate permutations in the two permutation classes and by the number of cycles each permutation
admits. We also refine this enumeration with respect to several statistics
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