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Register a new userOn permutations avoiding the dashed patterns 32-41 and 41-32
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David Callan
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We show that permutations of size $n$ avoiding both of the dashed patterns 32-41 and 41-32 are equinumerous with indecomposable set partitions of size $n+1$, and deduce a related result.
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An alternating permutation of length $n$ is a permutation $pi=pi_1 pi_2 ... pi_n$ such that $pi_1 < pi_2 > pi_3 < pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(sigma)$ be set of alternating permutations in $A_n$ that avoid a pattern $sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis.
Using the Herschel Space Observatorys Heterodyne Instrument for the Far-Infrared (HIFI), we have performed mapping observations of the 620.701 GHz 5(32)-4(41) transition of ortho-H2O within a roughly 1.5 x 1.5 arcmin region encompassing the Kleinmann-Low nebula in Orion, and pointed observations of that transition toward the Orion South condensation and the W49N region of high-mass star formation. Using the Effelsberg 100 m radio telescope, we obtained ancillary observations of the 22.23508 GHz 6(16)-5(23) water maser transition; in the case of Orion-KL, the 621 GHz and 22 GHz observations were carried out within 10 days of each other. The 621 GHz water line emission shows clear evidence for strong maser amplication in all three sources, exhibiting narrow (roughly 1 km/s FWHM) emission features that are coincident (kinematically and/or spatially) with observed 22 GHz features. Moreover, in the case of W49N - for which observations were available at three epochs spanning a two year period - the spectra exhibited variability. The observed 621 GHz/22 GHz line ratios are consistent with a maser pumping model in which the population
CONTEXT: Water vapour maser emission from evolved oxygen-rich stars remains poorly understood. Additional observations, including polarisation studies and simultaneous observation of different maser transitions may ultimately lead to greater insight. AIMS: We have aimed to elucidate the nature and structure of the VY CMa water vapour masers in part by observationally testing a theoretical prediction of the relative strengths of the 620.701 GHz and the 22.235 GHz maser components of ortho water vapour. METHODS: In its high-resolution mode (HRS) the Herschel Heterodyne Instrument for the Infrared (HIFI) offers a frequency resolution of 0.125 MHz, corresponding to a line-of-sight velocity of 0.06 km/s, which we employed to obtain the strength and linear polarisation of maser spikes in the spectrum of VY CMa at 620.701 GHz. Simultaneous ground based observations of the 22.235 GHz maser with the Max-Planck-Institut fur Radioastronomie 100-meter telescope at Effelsberg, provided a ratio of 620.701 GHz to 22.235 GHz emission. RESULTS:We report the first astronomical detection to date of water vapour maser emission at 620.701 GHz. In VY CMa both the 620.701 and the 22.235 GHz polarisation are weak. At 620.701 GHz the maser peaks are superposed on what appears to be a broad emission component, jointly ejected asymmetrically from the star. We observed the 620.701 GHz emission at two epochs 21 days apart, both to measure the potential direction of linearly polarised maser components and to obtain a measure of the longevity of these components. Although we do not detect significant polarisation levels in the core of the line, they rise up to approximately 6% in its wings.
We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.
A permutation $sigmainmathfrak{S}_n$ is simsun if for all $k$, the subword of $sigma$ restricted to ${1,...,k}$ does not have three consecutive decreasing elements. The permutation $sigma$ is double simsun if both $sigma$ and $sigma^{-1}$ are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.
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