575 research outputs found
Counting descents, rises, and levels, with prescribed first element, in words
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals
of Combinatorics, to appear 2007] refined the well-known permutation statistic
``descent'' by fixing parity of one of the descent's numbers. Results in that
paper were extended and generalized in several ways. In this paper, we shall
fix a set partition of the natural numbers , , and we study
the distribution of descents, levels, and rises according to whether the first
letter of the descent, rise, or level lies in over the set of words over
the alphabet . In particular, we refine and generalize some of the results
in [Counting occurrences of some subword patterns, Discrete Mathematics and
Theoretical Computer Science 6 (2003), 001-012.].Comment: 20 pages, sections 3 and 4 are adde
Place-difference-value patterns: A generalization of generalized permutation and word patterns
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson
introduced the notion of a "generalized permutation pattern" (GP) which
generalizes the concept of "classical" permutation pattern introduced by Knuth
in 1969. The invention of GPs led to a large number of publications related to
properties of these patterns in permutations and words. Since the work of
Babson and Steingrimsson, several further generalizations of permutation
patterns have appeared in the literature, each bringing a new set of
permutation or word pattern problems and often new connections with other
combinatorial objects and disciplines. For example, Bousquet-Melou et al.
introduced a new type of permutation pattern that allowed them to relate
permutation patterns theory to the theory of partially ordered sets.
In this paper we introduce yet another, more general definition of a pattern,
called place-difference-value patterns (PDVP) that covers all of the most
common definitions of permutation and/or word patterns that have occurred in
the literature. PDVPs provide many new ways to develop the theory of patterns
in permutations and words. We shall give several examples of PDVPs in both
permutations and words that cannot be described in terms of any other pattern
conditions that have been introduced previously. Finally, we raise several
bijective questions linking our patterns to other combinatorial objects.Comment: 18 pages, 2 figures, 1 tabl
A Cyclic Analogue of Stanley's Shuffle Theorem
We introduce the cyclic major index of a cycle permutation and give a
bivariate analogue of enumerative formula for the cyclic shuffles with a given
cyclic descent numbers due to Adin, Gessel, Reiner and Roichman, which can be
viewed as a cyclic analogue of Stanley's Shuffle Theorem. This gives an answer
to a question of Adin, Gessel, Reiner and Roichman, which has been posed by
Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.Comment: 7 pages. We thank Bruce Sagan for providing useful comments and
relevant references for the earlier versio
Retrievals of Cloud Droplet Size from the Research Scanning Polarimeter Data: Validation Using In Situ Measurements
We present comparisons of cloud droplet size distributions (DSDs) retrieved from the research scanning polarimeter (RSP) data with correlative in situ measurements made during the North Atlantic Aerosols and Marine Ecosystems Study (NAAMES). The airborne portion of this field experiment was based out of St. John's airport, Newfoundland, Canada with the focus of this paper being on the deployment in May - June 2016. RSP was onboard the NASA C-130 aircraft together with an array of in situ and other remote sensing instrumentation. The RSP is an along-track scanner measuring the polarized and total reflectance in 9 spectral channels. Its uniquely high angular resolution allows for characterization of liquid water droplet sizes using the rainbow structure observed in the polarized reflectance over the scattering angle range from 135 to 165.degrees The rainbow is dominated by single scattering of light by cloud droplets, so its structure is characteristic specifically of the droplet sizes at cloud top (within unit optical depth into the cloud, equivalent to approximately 50m). A parametric fitting algorithm applied to the polarized reflectance provides retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, the Rainbow Fourier Transform (RFT), which allows us to retrieve the droplet size distribution itself. The latter is important in the case of clouds with complex microphysical structure, or multiple layers of cloud, which result in multi-modal DSDs. During NAAMES the aircraft performed a number of flight patterns specifically designed for comparisons between remote sensing retrievals and in situ measurements. These patterns consisted of two flight segments above the same straight ground track. One of these segments was flown above clouds allowing for remote sensing measurements, while the other was near the cloud top where cloud droplets were sampled. We compare the DSDs retrieved from the RSP data with in situ measurements made by the Cloud Droplet Probe (CDP). The comparisons generally show good agreement (better than 1 micron for effective radius and in most cases better than 0.02 for effective variance) with deviations explainable by the position of the aircraft within the cloud, or by the presence of additional cloud layers between the cloud being sampled by the in situ instrumentation and the altitude of the remote sensing segment. In the latter case, the multi-modal DSDs retrieved from the RSP data were consistent with the multi-layer cloud structures observed in the correlative High Spectral Resolution Lidar (HSRL) profiles. The results of these comparisons provide a rare validation of polarimetric droplet size retrieval techniques, demonstrating their accuracy and robustness and the potential of satellite data of this kind on a global scale
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