Admissible pinnacle orderings

A pinnacle of a permutation is a value that is larger than its immediate neighbors when written in one-line notation. In this paper, we build on previous work that characterized admissible pinnacle sets of permutations. For these sets, there can be specific orderings of the pinnacles that are not admissible, meaning that they are not realized by any permutation. Here we characterize admissible orderings, using the relationship between a pinnacle x and its rank in the pinnacle set to bound the number of times that the pinnacles less than or equal to x can be interrupted by larger values.Comment: author added; error correcte

Pinnacle sets of signed permutations

Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been widely studied. In this article we define the pinnacle set of signed permutations of types $B$ and $D$. We give a closed formula for the number of type $B$/$D$ admissible pinnacle sets and answer several other related enumerative questions.Comment: 15 pages, 3 figures, to appear in Discrete Mathematic

Sociodemographic determinants of oral anticoagulant prescription in patients with atrial fibrillations: Findings from the PINNACLE registry using machine learning

Background: Current risk scores that are solely based on clinical factors have shown modest predictive ability for understanding of factors associated with gaps in real-world prescription of oral anticoagulation (OAC) in patients with atrial fibrillation (AF).Objective: In this study, we sought to identify the role of social and geographic determinants, beyond clinical factors associated with variation in OAC prescriptions using a large national registry of ambulatory patients with AF.Methods: Between January 2017 and June 2018, we identified patients with AF from the American College of Cardiology PINNACLE (Practice Innovation and Clinical Excellence) Registry. We examined associations between patient and site-of-care factors and prescription of OAC across U.S. counties. Several machine learning (ML) methods were used to identify factors associated with OAC prescription.Results: Among 864,339 patients with AF, 586,560 (68%) were prescribed OAC. County OAC prescription rates ranged from 26.8% to 93%, with higher OAC use in the Western United States. Supervised ML analysis in predicting likelihood of OAC prescriptions and identified a rank order of patient features associated with OAC prescription. In the ML models, in addition to clinical factors, medication use (aspirin, antihypertensives, antiarrhythmic agents, lipid modifying agents), and age, household income, clinic size, and U.S. region were among the most important predictors of an OAC prescription.Conclusion: In a contemporary, national cohort of patients with AF underuse of OAC remains high, with notable geographic variation. Our results demonstrated the role of several important demographic and socioeconomic factors in underutilization of OAC in patients with AF

Characteristics of the Mesophotic Megabenthic Assemblages of the Vercelli Seamount (North Tyrrhenian Sea)

The biodiversity of the megabenthic assemblages of the mesophotic zone of a Tyrrhenian seamount (Vercelli Seamount) is described using Remotely Operated Vehicle (ROV) video imaging from 100 m depth to the top of the mount around 61 m depth. This pinnacle hosts a rich coralligenous community characterized by three different assemblages: (i) the top shows a dense covering of the kelp Laminaria rodriguezii; (ii) the southern side biocoenosis is mainly dominated by the octocorals Paramuricea clavata and Eunicella cavolinii; while (iii) the northern side of the seamount assemblage is colonized by active filter-feeding organisms such as sponges (sometimes covering 100% of the surface) with numerous colonies of the ascidian Diazona violacea, and the polychaete Sabella pavonina. This study highlights, also for a Mediterranean seamount, the potential role of an isolated rocky peak penetrating the euphotic zone, to work as an aggregating structure, hosting abundant benthic communities dominated by suspension feeders, whose distribution may vary in accordance to the geomorphology of the area and the different local hydrodynamic conditions

Two-sided permutation statistics via symmetric functions

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi)$. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ whenever $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$ are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs, and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of $\operatorname{st}_{1}$ and $\operatorname{ist}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname{st}_{1}$ and $\operatorname{st}_{2}$. Our work leads to a rederivation of Stanley's generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and $\gamma$-positivity.Comment: 43 page

Fast hashing with Strong Concentration Bounds

Previous work on tabulation hashing by Patrascu and Thorup from STOC'11 on simple tabulation and from SODA'13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, e.g., the number of balls/keys hashing to a given bin, but under some quite severe restrictions on the expected values of these sums. The basic idea in tabulation hashing is to view a key as consisting of $c=O(1)$ characters, e.g., a 64-bit key as $c=8$ characters of 8-bits. The character domain $\Sigma$ should be small enough that character tables of size $|\Sigma|$ fit in fast cache. The schemes then use $O(1)$ tables of this size, so the space of tabulation hashing is $O(|\Sigma|)$. However, the concentration bounds by Patrascu and Thorup only apply if the expected sums are $\ll |\Sigma|$. To see the problem, consider the very simple case where we use tabulation hashing to throw $n$ balls into $m$ bins and want to analyse the number of balls in a given bin. With their concentration bounds, we are fine if $n=m$, for then the expected value is $1$. However, if $m=2$, as when tossing $n$ unbiased coins, the expected value $n/2$ is $\gg |\Sigma|$ for large data sets, e.g., data sets that do not fit in fast cache. To handle expectations that go beyond the limits of our small space, we need a much more advanced analysis of simple tabulation, plus a new tabulation technique that we call \emph{tabulation-permutation} hashing which is at most twice as slow as simple tabulation. No other hashing scheme of comparable speed offers similar Chernoff-style concentration bounds.Comment: 54 pages, 3 figures. An extended abstract appeared at the 52nd Annual ACM Symposium on Theory of Computing (STOC20

Species-Area Relationships of Cliff System Vegetational Communities in Cumberland Gap National Historical Park

A vegetational survey of vascular plants, bryophytes, and lichens was conducted on eleven discrete cliff systems in Cumberland Gap National Historical Park (CUGA) during the summer of 2011 and 2012. A total of 231 species were collected and identified, including 111 vascular plants, 37 bryophytes, and 83 lichens. Non-nested and nested species-area curves indicate that the sampling protocol was efficient at capturing diversity, and that larger cliff systems had higher levels of diversity. The steep slope of a generated log-transformed species-area curve, and the relatively low diversity compared to southern Appalachian forests, supports the hypothesis that cliff systems are similar to insular habitats. Multivariate analyses revealed that vascular plant, bryophyte, and lichen communities varied extensively across transects and cliff systems, largely unrelated to slope, aspect, or area. Based on these observations, it is imperative that each cliff site, and possibly each transect, be carefully surveyed before permitting recreational climbing. Several listed species were found during this survey, and all were established on smaller cliff systems. If the Park were most concerned with the protection of threatened, endangered, or disjunct species, it would be wise to preserve several small cliff systems, as opposed to a larger, more speciose cliff system