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Recursively defined combinatorial functions: extending Galton's board

DISCRETE MATHEMATICS 2001年第1-3期239卷 33-51页

作者： Neuwirth, E Univ Vienna OR&CS Dept Stat & Decis Support Syst A-1010 Vienna Austria

Many functions in combinatorics follow simple recursive relations of the type F(n,k) = a(n-1, k)F(n-1,k)+b(n-1,) Fk-1(n-1,k-1). Treating such functions as (infinite) triangular matrices and calling a(n,k) and b(n,k) generators of F, our paper will study the following question: Given two triangular arrays and their generators, how can we give explicit formulas for the generators of the product matrix? Our results can be applied to factor infinite matrices with specific types of generators (e.g. a(n,k) =a(n)' +a(k)(")) into matrices with 'simpler' types of generators. These factorization results then can be used to give construction methods for inverse matrices (yielding conditions for self-inverse matrices), and results for convolutions of recursively defined functions. Slightly extending the basic techniques, we will even be able to deal with certain cases of nontriangular infinite matrices. As a side-effect, many seemingly separate results about recursive combinatorial. functions will be shown to be special cases of the general framework developed here. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： combinatorial identities combinatorial functions triangular matrices inverse relations stirling numbers binomial coefficients

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A Recursive Basis for Primitive Forms in Symplectic Spaces and Applications to Heisenberg Groups

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Acta Mathematica Sinica,English Series 2016年第3期32卷 265-285页

作者： Annalisa BALDI Marilena BARNABEI Bruno FRANCHI Dipartimento di Matematica Piazza di Porta S.Donato 540126 BolognaItaly

This paper is divided in two parts： in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space（V^（2n）, ω）. Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H^n, n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin＇s complex of differential forms in H^n.

关键词： Symplectic manifolds differential forms Heisenberg groups combinatorial functions

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On the number of distinct multinomial coefficients

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JOURNAL OF NUMBER THEORY 2006年第1期118卷 15-30页

作者： Andrews, GE Knopfmacher, A Zimmermann, B Penn State Univ Dept Math University Pk PA 16802 USA Univ Witwatersrand John Knopfinacher Ctr Applicable Anal & Number Th Johannesburg South Africa Johannes Kepler Univ Linz Res Inst Symbol Computat A-4040 Linz Austria

We study M (n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both p(P)(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where p(P)(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials. (C) 2005 Elsevier Inc. All rights reserved.

关键词： factorials binomial coefficients combinatorial functions partitions of integers polynomial ideals Grobner bases

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Finite difference approach to trigonometry

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JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2011年第9期17卷 1365-1400页

作者： Antippa, Adel F. Univ Quebec Dept Phys Trois Rivieres PQ G9A 5H7 Canada

Two interrelated, finite difference and graph theoretic, approaches to trigonometry are developed by combining a generalization of the finite difference method first employed by Viete, with solution techniques, based on signal flow graphs, for finite difference equations with variable coefficients, and a scaling approach to trigonometry, based on the polygonometric identities.

关键词： exponential and trigonometric functions difference equations applications of graph theory factorials binomial coefficients combinatorial functions

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Coding Pulse Sequences Using a Combination of Factorial Pulse Coding and Arithmetic Coding

Coding Pulse Sequences Using a Combination of Factorial Puls...

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International Conference on Signal Processing and Communications

作者： Mittal, Udar Ramabadran, Tenkasi V. Ashley, James P. Motorola Inc Mobile Devices Business Bangalore 560093 Karnataka India Motorola Inc Mobile Devices Business Libertyville IL 60048 USA Motorola Inc Mobil Devices Business Schaumburg IL 60196 USA

ISBN: (纸本)9781424471379

A Factorial Pulse Coding approach using combinatorial functions for coding of pulse sequences is presented. An arithmetic coding approach for coding of pulse sequences is also described. Factorial pulse coding and arithmetic coding are compared. A method of combining the two approaches is proposed. The proposed combining method works by extending the pulse sequence by one bit whose probability is found from the arithmetic coding bounds.

关键词： arithmetic codes arithmetic coding coding pulse sequences combinatorial functions factorial pulse coding

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FLECK'S CONGRUENCE, ASSOCIATED MAGIC SQUARES AND A ZETA IDENTITY

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FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI 2011年第2期45卷 165-205页

作者： Lettington, Matthew C. Cardiff Univ Sch Math POB 926 Cardiff CF24 4YH S Glam Wales

Let the Fleck numbers, C-n(t, q), be defined such that C-n(t, q) = Sigma(k equivalent to q(mod n)) (-1)(k)((t)(k)) For prime p, Fleck obtained the result C-p(t, q) equivalent to 0(mod p(((Left perpendicular)/(t-1)/(p-1)(right perpendicular)))), where (Left perpendicular).(right perpendicular) denotes the usual floor function. This congruence was extended 64 years later by Weisman, in 1977, to include the case n = p(alpha) In this paper we show that the Fleck numbers occur naturally when one considers a symmetric n x n matrix, M, and its inverse under matrix multiplication. More specifically, we take M to be a symmetrically constructed n x n associated magic square of odd order, and then consider the reduced coefficients of the linear expansions of the entries of M-t with t is an element of Z. We also show that for any odd integer, n = 2m + 1, n >= 3, there exist geometric polynomials in m that are linked to the Fleck numbers via matrix algebra and p-adic interaction. These polynomials generate numbers that obey a reciprocal type of congruence to the one discovered by Fleck. As a by-product of our investigations we observe a new identity between values of the Zeta functions at even integers. Namely zeta(2j) = (-1)(j+1) (j pi(2)j/(2j + 1)! + Sigma(j=i)(k=1) (-1)(k)pi(2j-2k)/(2j - 2k + 1)! zeta(2k)) We conclude with examples of combinatorial congruences, Vandermonde type determinants and Number Walls that further highlight the symmetric relations that exist between the Fleck numbers and the geometric polynomials.

关键词： combinatorial identities combinatorial functions matrices determinants p-adic theory and binomial coefficients

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Expectations of functions of Sequences over Finite Alphabets with Given Transition Probabilities by Methods Independent of Sequence Length

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SIAM Journal on Computing 1972年第3期1卷 203-217页

作者： D. M. Jackson

A special case of the problem discussed in this paper occurs in connection with non-parametric classification and is introduced from this point of view. The special case concerns the computation of expectations of statistical functions of the “distance” between pairs of fixed-length sequences over a binary alphabet with given a priori state transition probabilities. The general problem involves an extension to alphabets of arbitrary order and the comparison of an arbitrary number of fixed-length sequences. Given a set of sequences, it is shown that for a large class of functions exact computation may be carried out by an algorithm whose computation time is independent of the length of the sequences. It is further shown that results for all functions of this class may be derived from a small number of basis functions. Two methods for computing basis functions are given. Basis functions for the commonly encountered special case involving pairs of binary sequences are given explicitly.

关键词： Nonparametric classification generating functions algorithms combinatorial functions

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