5,460 research outputs found
Distributions of several infinite families of mesh patterns
Br\"and\'en and Claesson introduced mesh patterns to provide explicit
expansions for certain permutation statistics as linear combinations of
(classical) permutation patterns. The first systematic study of avoidance of
mesh patterns was conducted by Hilmarsson et al., while the first systematic
study of the distribution of mesh patterns was conducted by the first two
authors.
In this paper, we provide far-reaching generalizations for 8 known
distribution results and 5 known avoidance results related to mesh patterns by
giving distribution or avoidance formulas for certain infinite families of mesh
patterns in terms of distribution or avoidance formulas for smaller patterns.
Moreover, as a corollary to a general result, we find the distribution of one
more mesh pattern of length 2.Comment: 27 page
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
Orbital Deflection of Comets by Directed Energy
Cometary impacts pose a long-term hazard to life on Earth. Impact mitigation
techniques have been studied extensively, but they tend to focus on asteroid
diversion. Typical asteroid interdiction schemes involve spacecraft physically
intercepting the target, a task feasible only for targets identified decades in
advance and in a narrow range of orbits---criteria unlikely to be satisfied by
a threatening comet. Comets, however, are naturally perturbed from purely
gravitational trajectories through solar heating of their surfaces which
activates sublimation-driven jets. Artificial heating of a comet, such as by a
laser, may supplement natural heating by the Sun to purposefully manipulate its
path and thereby avoid an impact. Deflection effectiveness depends on the
comet's heating response, which varies dramatically depending on factors
including nucleus size, orbit and dynamical history. These factors are
incorporated into a numerical orbital model to assess the effectiveness and
feasibility of using high-powered laser arrays in Earth orbit and on the ground
for comet deflection. Simulation results suggest that a diffraction-limited 500
m orbital or terrestrial laser array operating at 10 GW for 1% of each day over
1 yr is sufficient to fully avert the impact of a typical 500 m diameter comet
with primary nongravitational parameter A1 = 2 x 10^-8 au d^-2. Strategies to
avoid comet fragmentation during deflection are also discussed.Comment: 13 pages, 12 figures; AJ, in pres
Avoiding vincular patterns on alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). The study of alternating words avoiding classical permutation
patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it
was shown that 123-avoiding up-down words of even length are counted by the
Narayana numbers.
However, not much was understood on the structure of 123-avoiding up-down
words. In this paper, we fill in this gap by introducing the notion of a
cut-pair that allows us to subdivide the set of words in question into
equivalence classes. We provide a combinatorial argument to show that the
number of equivalence classes is given by the Catalan numbers, which induces an
alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.
Further, we extend the enumerative results in~\cite{GKZ} to the case of
alternating words avoiding a vincular pattern of length 3. We show that it is
sufficient to enumerate up-down words of even length avoiding the consecutive
pattern and up-down words of odd length avoiding the
consecutive pattern to answer all of our enumerative
questions. The former of the two key cases is enumerated by the Stirling
numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic
Pattern-avoiding alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). In this paper, we initiate the study of (pattern-avoiding)
alternating words. We enumerate up-down (equivalently, down-up) words via
finding a bijection with order ideals of a certain poset. Further, we show that
the number of 123-avoiding up-down words of even length is given by the
Narayana numbers, which is also the case, shown by us bijectively, with
132-avoiding up-down words of even length. We also give formulas for
enumerating all other cases of avoidance of a permutation pattern of length 3
on alternating words
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