An Asymptotic Distribution Theory for Eulerian Recurrences with Applications
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Counting Prime Polynomials and Measuring Complexity and Similarity of Information
Counting Prime Polynomials and Measuring Complexity and Similarity of Information by Niko Rebenich B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012 A Dissertation Submitted in Partial Fulllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Electrical and Computer Engineering © Niko Rebenich, 2016 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Counting Prime Polynomials and Measuring Complexity and Similarity of Information by Niko Rebenich B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012 Supervisory Committee Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering) Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering) Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science) iii Supervisory Committee Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering) Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering) Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science) ABSTRACT This dissertation explores an analogue of the prime number theorem for polynomi- als over nite elds as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational eort. In this context asymptotic series expan- sions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. -
In Combinatorics, the Eulerian Number A(N, M), Is the Number of Permutations of the Numbers 1 to N in Which Exactly M Elements A
From Wikipedia, the free encyclopedia In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). They are the coefficients of the Eulerian polynomials: The Eulerian polynomials are defined by the exponential generating function The Eulerian polynomials can be computed by the recurrence An equivalent way to write this definition is to set the Eulerian polynomials inductively by Other notations for A(n, m) are E(n, m) and . 1History 2 Basic properties 3 Explicit formula 4 Summation properties 5 Identities 6 Eulerian numbers of the second kind 7 References 8 External links In 1755 Leonhard Euler investigated in his book Institutiones calculi differentialis polynomials α1(x) = 1, α2(x) = x + 1, 2 α3(x) = x + 4x + 1, etc. (see the facsimile). These polynomials are a shifted form of what are now called the Eulerian polynomials An(x). For a given value of n > 0, the index m in A(n, m) can take values from 0 to n − 1. For fixed n there is a single permutation which has 0 ascents; this is the falling permutation (n, n − 1, n − 2, ..., 1). There is also a single permutation which has n − 1 ascents; this is the rising permutation (1, 2, 3, ..., n). Therefore A(n, 0) and A(n, n − 1) are 1 for all values of n. Reversing a permutation with m ascents creates another permutation in which there are n − m − 1 ascents. Therefore A(n, m) = A(n, n − m − 1). -
A Unified Approach to Combinatorial Triangles: a Generalized Eulerian Polynomial
A unified approach to combinatorial triangles: a generalized Eulerian polynomial Bao-Xuan Zhu∗ School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China Abstract Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array [ n,k]n,k 0 satisfying the recurrence relation: T ≥ cd n,k = λ(a0n + a1k + a2) n 1,k + (b0n + b1k + b2) n 1,k 1 + (n k + 1) n 1,k 2 T T − T − − λ − T − − with = 1 and = 0 unless 0 k n. We derive a functional transformation T0,0 Tn,k ≤ ≤ for its row-generating function (x) from the row-generating function A (x) of Tn n another array [An,k]n,k satisfying a two-term recurrence relation. Based on this transformation, we can get properties of and (x) including nonnegativity, Tn,k Tn log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangu- lar array from staircase tableaux and the alternating-runs triangle of type B in a unified approach. arXiv:2007.12602v1 [math.CO] 24 Jul 2020 MSC: 05A15; 05A19; 05A20; 26C10 Keywords: Eulerian numbers; Eulerian polynomials; Recurrence relations; Gener- ating functions; Real zeros; Log-concavity; Strong q-log-convexity; γ-positivity; 1 Introduction 1.1 Eulerian numbers Let Sn denote the symmetric group on n-elements [n]= 1, 2,...,n . -
Factorial Stirling Matrix and Related Combinatorial Sequences
Linear Algebra and its Applications 357 (2002) 247–258 www.elsevier.com/locate/laa Factorial Stirling matrix and related combinatorial sequencesୋ Gi-Sang Cheon a,∗, Jin-Soo Kim b aDepartment of Mathematics, Daejin University, Pocheon 487-711, Republic of Korea bDepartment of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 21 November 2001; accepted 9 April 2002 Submitted by R.A. Brualdi Abstract In this paper, some relationships between the Stirling matrix, the Vandermonde matrix, the Benoulli numbers and the Eulerian numbers are studied from a matrix representation of k!S(n,k) which will be called the factorial Stirling matrix, where S(n,k) are the Stirling numbers of the second kind. As a consequence a number of interesting and useful identities are obtained. © 2002 Elsevier Science Inc. All rights reserved. AMS classification: 05A19; 05A10 Keywords: Stirling number; Bernoulli number; Eulerian number 1. Introduction In many contexts (see [1–4]), a number of interesting and useful identities involv- ing binomial coefficients are obtained from a matrix representation of a particular counting sequence. Such a matrix representation provides a powerful computational tool for deriving identities and an explicit formula related to the sequence. First, we begin our study with a well-known definition: for any real number x and an integer k,define ୋ This work was supported by the Daejin University Research Grants in 2001. ∗ Corresponding author. E-mail addresses: [email protected] (G.-S. Cheon), [email protected] (J.-S. Kim). 0024-3795/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. -
Bernoulli and Euler Polynomials
Bernoulli and Euler Polynomials Güneş Çatma Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in Mathematics Eastern Mediterranean University July 2013 Gazimağusa, North Cyprus Approval of the Institute of Graduate Studies and Research Prof. Dr. Elvan Yılmaz Director I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Mathematics. Prof. Dr. Nazım Mahmudov Chair, Department of Mathematics We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Mathematics. Assoc. Prof. Dr. Sonuç Zorlu Oğurlu Supervisor Examining Committee 1. Prof. Dr. Nazım Mahmudov 2. Assoc. Prof. Dr. Mehmet Ali Özarslan 3. Assoc. Prof. Dr. Sonuç Zorlu Oğurlu ABSTRACT This thesis provides an overview of Bernoulli and Euler numbers. It describes the Bernoulli and Euler polynomials and investigates the relationship between the classes of the two polynomials. It also discusses some important identities using the finite difference calculus and differentiation. The last part of this study is con- cerned with the Generalized Bernoulli and Euler polynomials. Furthermore, the properties obtained in the second chapter are also examined for the generalized Bernoulli and Euler polynomials in this part of the thesis. The Complemen- tary Argument Theorem, the generating functions, the Multiplication and the Euler-Maclauren Theorems are widely used in obtaining the mentioned results. Keywords: Bernoulli -Euler Polynomials, Generalized Bernoulli -Euler Polyno- mials, Finite Difference iii OZ¨ Bu ¸calı¸smadaBernoulli ve Euler sayıları ile Bernoulli ve Euler polinom- ları arasındaki ili¸skilerincelenmi¸stir. -
Generalized Eulerian Numbers
JOURNAL OF COMBINATORIALTHEORY, Series A 32, 195-215 (1982) Generalized Eulerian Numbers D. H. LEHMER Department of Mathematics, University of Cal$ornia, Berkeley, California 94720 Received March 17, 1981 DEDICATED TO PROFESSOR MARSHALL HALL, JR., ON THE OCCASION OF HIS RETIREMENT 1. INTRODUCTION The Eulerian numbers (not to be confused with the Euler numbers E,) were discovered by Euler in 1755 [I]. Since then, they have been rediscovered, redefined, generalized and used by many authors [2-151. Explicitly they can be defined for 0 < k < n by k-l A(n,k)= 1 (-1)’ (k -j)“. j=O Euler wrote g+ f z-z,(x) P/n! n=O and (x - 1)” H,(x) = i A@, k) xk-‘. &=I This can be expressed by the generating function l-u = 1 + 2 f A@, k) ukt”/n! 1 - u exp{t(l - u)} &=I n=l from which the recurrence relation A@+ l,k)=(n-k+2)A(n,k- l)+kA(k,t) (3) is easily derived (see, for example, Comtet [ 15, Vol. 1, pp. 63-641. 195 0097-3 165/82/020195-21$02.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. 196 D. H. LEHMER The connections between A(n, k) and the polynomials of Bernoulli and Euler, such as ml(x)-B,(O)) = ;:r (X+;-l)A(n-l,k), were among the first discoveries made. Later Eulerian Numbers began to occur in combinatorial problems and A@, k) is best remembered as the number of permutations 1 2 )...) n $l), 742) ,***,70) that have precisely k - 1 “rises,” where a rise is a pair of consecutive marks n(i), n(i + 1) with n(i) < n(i + 1). -
General Eulerian Polynomials As Moments Using Exponential Riordan Arrays
1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.9.6 47 6 23 11 General Eulerian Polynomials as Moments Using Exponential Riordan Arrays Paul Barry School of Science Waterford Institute of Technology Ireland [email protected] Abstract Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the general Eulerian polynomials, as defined by Xiong, Tsao and Hall, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of this polynomial sequence in terms of continued fractions, and we also calculate the Hankel transforms of the polynomial sequence. We indicate that the polynomial sequence can be characterized by the further notion of generalized Eulerian distribution first introduced by Morisita. We finish with examples of related Pascal-like triangles. 1 Introduction The triangle of Eulerian numbers 10 0 000 ... 10 0 000 ... 11 0 000 ... 14 1 000 ... , 1 11 11 1 0 0 ... 1 26 66 26 1 0 ... . . .. 1 with its general elements k n−k n +1 n +1 A = (−1)i (k − i + 1)n = (−1)i (n − k − i)n, n,k i i i=0 i=0 X X along with its variants, has been studied extensively [1, 10, 14, 15, 17, 18]. It is closely associated with the family of Eulerian polynomials k En(t)= An,kt . k X=0 The Eulerian polynomials have exponential generating function ∞ xn t − 1 E (t) = . n n! t − ex(t−1) n=0 X It can be shown that the sequence En(t) is the moment sequence of a family of orthogonal polynomials [3]. -
Eulerian Polynomials and B-Splines
Eulerian Polynomials and B-Splines Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 61702-2900, USA Abstract Here presented is the interrelationship between Eulerian polynomials, Eule- rian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, B- splines, respectively. The properties of Eulerian polynomials and Eulerian frac- tions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed. AMS Subject Classification: 41A80, 65B10, 33C45, 39A70. Key Words and Phrases: B-spline, Eulerian polynomial, Eulerian Number, Eulerian Fraction, exponential Euler polynomial, Euler-Frobenius polynomial, and Euler-Frobenius fractions. 1 Introduction Eulerian polynomial sequence fAn(z)gn≥0 is defined by the following summation (cf. for examples, [1], [2], and [3]): X An(z) `nz` = ; jzj < 1: (1.1) (1 − z)n+1 `≥0 It is well-known that the Eulerian polynomial, An(z) (see p. 244 of [3]), of degree n can be written in the form n X k An(z) = A(n; k)z ;A0(z) = 1; (1.2) k=1 where A(n; k) are called the Eulerian numbers that can be calculated by using k X n + 1 A(n; k) = (−1)j (k − j)n; 1 ≤ k ≤ n; (1.3) j j=0 1 2 T. X. He and A(n; 0) = 1. The Eulerian numbers are found, for example, in the standard treatises by Riordan [4], Comtet [3], Aigner [5], Grahm et al. [6], etc. However, their n notations are not standard, which are shown by A , A(n; k), W , , and n;k n;k k E(n; k), respectively. -
Complexity Theory and Its Applications in Linear Quantum
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2016 Complexity Theory and its Applications in Linear Quantum Optics Jonathan Olson Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Physical Sciences and Mathematics Commons Recommended Citation Olson, Jonathan, "Complexity Theory and its Applications in Linear Quantum Optics" (2016). LSU Doctoral Dissertations. 2302. https://digitalcommons.lsu.edu/gradschool_dissertations/2302 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. COMPLEXITY THEORY AND ITS APPLICATIONS IN LINEAR QUANTUM OPTICS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Jonathan P. Olson M.S., University of Idaho, 2012 August 2016 Acknowledgments My advisor, Jonathan Dowling, is apt to say, \those who take my take my advice do well, and those who don't do less well." I always took his advice (sometimes even against my own judgement) and I find myself doing well. He talked me out of a high-paying, boring career, and for that I owe him a debt I will never be able to adequately repay. My mentor, Mark Wilde, inspired me to work hard without saying a word about what I \should" be doing, and instead leading by example. -
Generalized J-Factorial Functions, Polynomials, and Applications
1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.6.7 47 6 23 11 Generalized j-Factorial Functions, Polynomials, and Applications Maxie D. Schmidt University of Illinois, Urbana-Champaign Urbana, IL 61801 USA [email protected] Abstract The paper generalizes the traditional single factorial function to integer-valued mul- tiple factorial (j-factorial) forms. The generalized factorial functions are defined recur- sively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial functions. The resulting coefficient triangles are similar to the classical sets of Stirling numbers and satisfy many analogous finite-difference and enumerative properties as the well-known combinatorial triangles. The generalized triangles are also considered in terms of their relation to elementary symmetric poly- nomials and the resulting symmetric polynomial index transformations. The definition of the Stirling convolution polynomial sequence is generalized in order to enumerate the parametrized sets of j-factorial polynomials and to derive extended properties of the j-factorial function expansions. The generalized j-factorial polynomial sequences considered lead to applications expressing key forms of the j-factorial functions in terms of arbitrary partitions of the j-factorial function expansion triangle indices, including several identities related to the polynomial expansions of binomial coefficients. Additional applications include the formulation of closed-form identities and generating functions for the Stirling numbers of the first kind and r-order harmonic number sequences, as well as an extension of Stirling’s approximation for the single factorial function to approximate the more general j-factorial function forms. 1 Notational Conventions Donald E. -
Generalized Euler Sums
Generalized Eulerian Sums Fan Chung∗ Ron Graham∗ Dedicated to Adriano Garsia on the occasion of his 84th birthday. Abstract In this note, we derive a number of symmetrical sums involving Eu- lerian numbers and some of their generalizations. These extend earlier identities of Don Knuth and the authors, and also include several q-nomial sums inspired by recent work of Shareshian and Wachs on the joint distri- bution of various permutation statistics, such as the number of excedances, the major index and the number of fixed points of a permutation. We also produce symmetrical sums involving \restricted" Eulerian numbers which count permutations π on f1; 2; : : : ; ng with a given number of descents and which, in addition, have the value of π−1(n) specified. 1 Introduction n The classical Eulerian numbers, which we will denote by k , occur in a variety of contexts in combinatorics, number theory and computer science (see [4]). These numbers were introduced by Euler [3] in 1736 and have many interesting properties. We list some small values in Table 1. k 0 1 2 3 4 5 1 1 0 0 0 0 0 2 1 1 0 0 0 0 n 3 1 4 1 0 0 0 4 1 11 11 1 0 0 5 1 26 66 26 1 0 6 1 57 302 302 57 1 n Some Eulerian numbers k Table 1 n In particular, k enumerates the number of permutations π on [n] := f1; 2; : : : ; ng which have k descents (i.e., i ≤ n−1 with π(i) > π(i+1) ) as well as the number ∗University of California, San Diego 1 of permutations π on [n] which have k excedances (i.e., i < n with π(i) > i). -
Haystack Hunting Hints and Locker Room Communication∗
Haystack Hunting Hints and Locker Room Communication∗ Artur Czumaj † George Kontogeorgiou ‡ Mike Paterson § Abstract We want to efficiently find a specific object in a large unstructured set, which we model by a random n-permutation, and we have to do it by revealing just a single element. Clearly, without any help this task is hopeless and the best one can do is to select the element at random, and 1 achieve the success probability n . Can we do better with some small amount of advice about the permutation, even without knowing the object sought? We show that by providing advice of just one integer in 0; 1; : : : ; n 1 , one can improve the success probability considerably, by log n f − g a Θ( log log n ) factor. We study this and related problems, and show asymptotically matching upper and lower bounds for their optimal probability of success. Our analysis relies on a close relationship of such problems to some intrinsic properties of random permutations related to the rencontres number. 1 Introduction Understanding basic properties of random permutations is an important concern in modern data science. For example, a preliminary step in the analysis of a very large data set presented in an unstructured way is often to model it assuming the data is presented in a random order. Un- derstanding properties of random permutations would guide the processing of this data and its analysis. In this paper, we consider a very natural problem in this setting. You are given a set of 1 n objects ([n 1], say ) stored in locations x ; : : : ; x − according to a random permutation σ of − 0 n 1 [n 1].