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 $q$-Eulerian polynomials for Weyl groups of type $D$, 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 $q$-Eulerian polynomials of type $D$, 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 $q$, and hence prove Brenti's conjecture. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, 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 $D$. 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 $D$. 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 $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (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 $\underline{132}$ and up-down words of odd length avoiding the consecutive pattern $\underline{312}$ 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 $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (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