Number Of Descents Research Articles

Hindu Avatāra and Christian Kenosis: A New Approach in Comparative Theology1

Abstract Within comparative theology, the Hindu doctrine of avatāra has traditionally been compared to the Christian doctrine of the incarnation, both of which are expressions of divine embodiment in creaturely form. This article, however, contrasts the doctrine of avatāra with a reading of the incarnation that frames the latter within the broader context of kenoticism developed primarily in modern Christian theology. I argue that this approach (a) alleviates some of the difficulties encountered by the traditional comparison of avatāra with the incarnation and (b) highlights some noteworthy but easily missed dimensions present in the doctrines of both faith traditions. This is established through a comparison of two stories of young Kṛṣṇa and their display of divine play, līlā, with Hans Urs von Balthasar's kenotic reading of both the intratrinitarian relations and the metaphysical structure of finite beings. In this way, the article, on the one hand, alleviates the discrepancies about the number of divine descents and the link between these and the world's creation and metaphysical constitution, and, on the other hand, highlights how the Hindu notion of līlā and Christian kenotic readings intend to express the dialectic of divine sovereignty and accessibility in a philosophically and theologically responsible way.

Moderate deviations for the number of descents in a random permutation

Moderate deviations for the number of descents in a random permutation

Sharp analysis on the joint distribution of the number of descents and inverse descents in a random permutation

Sharp analysis on the joint distribution of the number of descents and inverse descents in a random permutation

Discretized Normal Approximation for the Number of Descents

This paper explores the probability approximation of the number of descents in a random permutation. It is known that the distribution of the number of descents can be approximated by a normal distribution under Kolmogorov distance. Subsequently, an explicit constant is provided in the bound of 13.42. The objective is to prove a similar result but using a stronger distance, namely, the total variation distance. The Stein’s method and the exchangeable pair transformation has been used to give a bound for discretized normal approximation for the distribution of number of descents under the total variation distance. The result obtained gave an improved constant of 10.71.

Is the six-minute step test an alternative to six-minute walking test to assessment the functional capacity in elderly?

Abstract Background The assessment of cardiorespiratory functional capacity can contribute to the prognosis of cardiovascular, chronic respiratory and musculoskeletal system diseases, considering loss of functional capacity and the presence of comorbidities. This assessment can be done through maximal tests such as the cardiopulmonary exercise test, exercise test and submaximal tests such as the 6-minute walk test (6MWT) and the 6-minute step test (6MST). There are few studies that compare both tests and can validate the replacement of the 6MWT by the 6MST, since the 6MWT is a test that requires a large space to be performed, unlike the 6MST, which only needs a 20 cm high step. Purpose The objective of the present study was to verify the correlation between the 6MST and the 6MWT in the evaluation of the functional capacity of elderly people. Methods Cross-sectional descriptive study with voluntary non-probabilistic sampling, consisting of 30elderly people who are part of the Cardiovascular Physiotherapy in Primary Care study. The sample was submitted to the 6MWT and 6MST, and the beginning and end parameters were collected to be compared. Submaximal exercise capacity was evaluated by the distance covered in the walking test and by the number of steps taken up and down, and equations were performed to obtain the VO2max of both tests. Results The sample was predominantly female (75%), with an average age of 69.86 (±7.34) years. There was a positive and moderate correlation between the estimated values for VO2max in the walk and step test (p = 0.01 and r = 0.57) and between the distance covered in the walk test and the number of ascents and descents in the step test (p = 0.02 and r = 0.43). Conclusion The six-minute step test can be an alternative to assess the functional capacity of the elderly.

The total number of descents and levels in (cyclic) tensor words

The total number of descents and levels in (cyclic) tensor words

Sharp large deviations and concentration inequalities for the number of descents in a random permutation

Abstract The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin–Hall distribution, we prove that the number of descents satisfies a sharp large-deviation principle. A very precise concentration inequality involving the rate function in the large-deviation principle is also provided.

Two-sided permutation statistics via symmetric functions

Abstract Given a permutation statistic $\operatorname {\mathrm {st}}$ , define its inverse statistic $\operatorname {\mathrm {ist}}$ by . We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {ist}}_{2}$ whenever $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {st}}_{2}$ 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 $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {ist}}_{2}$ can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of $\operatorname {\mathrm {st}}_{1}$ and $\operatorname {\mathrm {st}}_{2}$ . 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.

Non-overlapping descents and ascents in stack-sortable permutations

Non-overlapping descents and ascents in stack-sortable permutations

Analisis Kecelakaan Lalu Lintas Ruas Jalan Padang - Solok dengan Metode Accident Rate

The Padang-Solok road section is one of the national road sections that frequently experiences accidents every year. The aim of this research is to determine the factors that cause accidents on the Jalan Lubuk Paraku Sitinjau Lauik section based on the accident rate, and find out alternative prevention solutions future traffic accidents. This type of research is descriptive research using the Accident Rate method. The factors reviewed are the total number of accidents that occur and the length of the road section studied. An Accident Rate value > 1.0 will be classified as a Black Spot, while a value < 1.0 will be classified as a Black Site. The results of the accident analysis based on the Accident Rate can be identified in three road sections that show a high accident rate, namely more than 1.0. The highest accident rate was on the Panorama 2 – Padang-Solok City Limits road section at 2.37 accidents/km.year. The most dominant factor causing traffic accidents is the human factor. After an inspection was carried out, the brakes failed due to the large number of descents or steep inclines which made the car brakes not function properly. An alternative solution for preventing accidents is to create a temporary stopping place at the Padang - Solok city limits to cool down the engine and car brakes, repair traffic signs that are not yet permanent and create a special brake rescue lane for broken brakes at Panorama 2 and near the rice fields PU Mes.

Descents on nonnesting multipermutations

Descents on nonnesting multipermutations

Consecutive patterns in Catalan words and the descent distribution

In this paper, we compute the distribution of the descent statistic on the Catalan words avoiding a consecutive pattern of length at most three. Baril–Kirgizov–Vajnovszki began the study of patterns in Catalan words, focusing on the enumeration of those that avoid classical patterns of length 3. We make use of the symbolic method to associate functional equations satisfied by the counting generating functions. As a consequence we enumerate the set of Catalan words that avoid consecutive patterns of length 3, and we also provide the total number of descents on this set.

Descent distribution on Catalan words avoiding ordered pairs of relations

Descent distribution on Catalan words avoiding ordered pairs of relations

Symmetric generating functions and Euler–Stirling statistics on permutations

Symmetric generating functions and Euler–Stirling statistics on permutations

Is the six-minute step test able to reflect the severity and symptoms based on cat score?

Is the six-minute step test able to reflect the severity and symptoms based on cat score?

A Cyclic Analogue of Stanley's Shuffling Theorem

We introduce the cyclic major index of a cyclic permutation and give a bivariate analogue of the enumerative formula for the cyclic shuffles with a given cyclic descent number due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's shuffling 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.

The bipermutahedron

The harmonic polytope and the bipermutahedron are two related polytopes that arose in the Lagrangian geometry of matroids. We study the bipermutahedron. We show that it is a simple polytope whose faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its \(f\)-vector is a simple evaluation of the three variable Rogers-Ramanujan function. We introduce the biEulerian polynomial, which counts bipermutations according to their number of descents, and equals the \(h\)-polynomial of the bipermutahedral fan. We construct a unimodular triangulation of the product \(\Delta \times \cdots \times \Delta\) of triangles that is combinatorially equivalent to (the triple cone over) the bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the \(h\)-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.Mathematics Subject Classifications: 52B20, 52B05, 05A15Keywords: Polytope, bipermutahedron, bipermutations, descents, \(f\)-vector, \(h\)-vector, unimodular triangulation, Ehrhart polynomial, real-rooted polynomial, deformation cone

Quasi-Stirling polynomials on multisets

Quasi-Stirling polynomials on multisets

On a Conjecture Concerning Shuffle-Compatible Permutation Statistics

The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of the shuffle compatibility property of permutation statistics.

A refinement of the Murnaghan-Nakayama rule by descents for border strip tableaux

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer (1974) and James & Kerber (1984) imply that, mysteriously, its evaluation at a \(k\)-th primitive root of unity yields the number of border strip tableaux with all strips of size \(k\), up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new statistic for border strip tableaux, extending the classical definition of descents in standard Young tableaux. Curiously, it turns out that our new statistic is very closely related to a descent set for tuples of standard Young tableaux appearing in the quasisymmetric expansion of LLT polynomials given by Haglund, Haiman and Loehr (2005).Mathematics Subject Classifications: 05A19, 05E10Keywords: Border strip tableaux, descents, Murnaghan-Nakayama rule, fake degree