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تسجيل مستخدم جديدOn 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.
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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.
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