scholarly journals An Interesting New Mahonian Permutation Statistic

10.37236/419 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mark C. Wilson
Keyword(s):  
Joint Distribution ◽  
Random Permutation ◽  
General Construction ◽  
Standard Algorithm ◽  
Permutation Statistic

The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic DIS that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor does its joint distribution with the standard Eulerian statistics des and exc appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner eul such that (eul, DIS) is equidistributed with (des, MAJ). However eul itself appears not to be a known Eulerian statistic. Several ideas for further research on this topic are listed.

scholarly journals Local extrema in random permutations and the structure of longest alternating subsequences

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Dan Romik
Keyword(s):  
Joint Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Local Minima ◽  
New Approach ◽  
Successive Minimum ◽  
Local Extrema ◽  
Nous Montrons ◽  
Dimensional Distribution ◽  
International Audience

International audience Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$. Pour une permutation aléatoire d'ordre $n$, on désigne par $\textbf{as}_n$ la longueur maximale d'une de ses sous-suites alternantes. Stanley a étudié la distribution de $\textbf{as}_n$ en utilisant des méthodes algébriques, et il a démontré en particulier que $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ et $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. A partir du résultat de Stanley on peut montrer qu'après changement d'échelle, $\textbf{as}_n$ converge vers la distribution normale. Nous présentons ici une approche nouvelle pour l'étude de $\textbf{as}_n$, en la reliant à la suite des extrema locaux d'une permutation aléatoire, dont nous montrons qu'elle constitue une sous-suite alternante maximale "canonique''. En utilisant cette relation, nous prouvons à nouveau les résultats mentionnés ci-dessus d'une façon plus probabiliste et transparente. En plus, nous prouvons un résultat asymptotique sur la distribution limite des paires formées d'un minimum et d'un maximum locaux consécutifs.

scholarly journals On random polynomials over finite fields

1993 ◽  
Vol 114 (2) ◽  
pp. 347-368 ◽  
Author(s):  
Richard Arratia ◽  
A. D. Barbour ◽  
Simon Tavaré
Keyword(s):  
Limit Theorem ◽  
Joint Distribution ◽  
Negative Binomial ◽  
Random Permutation ◽  
Discrete Distributions ◽  
Random Polynomials ◽  
Discrete Approximations ◽  
Number Of Factors ◽  
Variation Distance ◽  
Functional Central Limit

AbstractWe consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

Emanuel Ringelblum przed wojną: człowiek i historyk

2007 ◽  
pp. 211-220
Author(s):  
Samuel Kassow
Keyword(s):  
Joint Distribution ◽  
Education Agency ◽  
Active Membership

This article discusses the pre-war life of Emanuel Ringelblum – from the organisation of the Junger Historiker Krajz (the circle of young Jewish historians) at Warsaw University, through his YIVO activity, his involvement in the setting up of tourist associations, work for the Joint Distribution Committee as editor-in-chief of „Folkshilf”, active membership in Poale Zion-Left (he ran its most important education agency: the Ovnt kursn far arbiter) to his involvement in organisation of aid for Jews in the transit camp in Zbąszyń in 1938.

ФІГУРА УСЛАВЛЕННЯ ЯК ТЕМБРАЛЬНЕ НАПОВНЕННЯ ПАРТІЇ КНЯЗЯ ІГОРЯ В ОДНОЙМЕННІЙ ОПЕРІ О. БОРОДІНА

2020 ◽  
Author(s):  
Admink Admink
Keyword(s):  
Cultural Studies ◽  
General Construction ◽  
Modern Sense

Продемонстровано уявлення в сучасному українському мистецтвознавстві та культурології відомостей про тембральну стратегію й загальну конструкцію опери «Князь Ігор» за авторськими установками у науковому передбаченні сучасного смислу розуміння подій «Слова о полку Ігоревім» в композиції твору О. Бородіна.Ключові слова: славлення, тембр співу, опера, семантика баса, «Князь Ігор» О. Бородіна. The article demonstrates the presentation in Ukrainian art and cultural studies of information about the timbre strategy and the general construction of the opera Prince Igor in accordance with the author's attitudes and scientific foresight of the modern sense of understanding the events of the Word of Igor's Campaign in the composition of A. Borodin.Key words: glorification, timbre of singing, opera, bass semantics, «Prince Igor» by A. Borodin

Sign in / Sign up

Export Citation Format

Share Document