Moa Apagodu, Doron Zeilberger,
... William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence ...
Tópico(s): Coding theory and cryptography
2017 - Taylor & Francis | American Mathematical Monthly
Manuel Kauers, Doron Zeilberger,
... Foundation. Additional informationNotes on contributorsManuel Kauers 1 2 Doron Zeilberger 3
Tópico(s): Advanced Algebra and Geometry
2008 - Taylor & Francis | The Journal of Difference Equations and Applications
Brian Nakamura, Doron Zeilberger,
... avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a ...
Tópico(s): History and advancements in chemistry
2012 - Elsevier BV | Advances in Applied Mathematics
Moa Apagodu, Doron Zeilberger,
Algorithms for multi-sum summation and intergration of hypergeometric summands and integrands are given and sharp upper bounds for the orders are presented.
Tópico(s): Polynomial and algebraic computation
2006 - Elsevier BV | Advances in Applied Mathematics
Mohamud Mohammed, Doron Zeilberger,
Tópico(s): Coding theory and cryptography
2004 - Elsevier BV | Journal of Symbolic Computation
William Y. C. Chen, Qing-Hu Hou, Doron Zeilberger,
Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of P(x)kQ(x), for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by ak, we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum ∑k=0rp-1ak, for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant ...
Tópico(s): Advanced Combinatorial Mathematics
2016 - Taylor & Francis | The Journal of Difference Equations and Applications
Svante Janson, Brian Nakamura, Doron Zeilberger,
We study statistical properties of the random variables X σ (π), the number of occurrences of the pattern σ in the permutation π.We present two contrasting approaches to this problem: traditional probability theory and the "less traditional" computational approach.Through the perspective of the first approach, we prove that for any pair of patterns σ and τ , the random variables X σ and X τ are jointly asymptotically normal (when the permutation is chosen from S n ).From the other perspective, we develop ...
Tópico(s): Stochastic processes and statistical mechanics
2015 - Electronic Journal of Combinatorics | Journal of Combinatorics
Christoph Koutschan, Thotsaporn Thanatipanonda,
... algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger's ...
Tópico(s): Mathematical Dynamics and Fractals
2013 - Birkhäuser | Annals of Combinatorics
Tópico(s): Coding theory and cryptography
2012 - Springer Science+Business Media | The Ramanujan Journal
Andrew V. Sills, Doron Zeilberger,
The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of "rigorous guessing" as facilitated by the quasi-polynomial ansatz.
Tópico(s): Analytic Number Theory Research
2012 - Elsevier BV | Advances in Applied Mathematics
Christoph Koutschan, Manuel Kauers, Doron Zeilberger,
The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.
Tópico(s): Advanced Mathematical Identities
2011 - National Academy of Sciences | Proceedings of the National Academy of Sciences
E. Rodney Canfield, Svante Janson, Doron Zeilberger,
The Mahonian statistic is the number of inversions in a permutation of a multiset with a i elements of type i , 1 ⩽ i ⩽ m . The counting function for this statistic is the q analog of the multinomial coefficient ( a 1 + ⋯ + a m a 1 , … , a m ) , and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is computer-assisted , based on the method of moments. The Maple package MahonianStat , available from the webpage of this article, ...
Tópico(s): Mathematical Dynamics and Fractals
2010 - Elsevier BV | Advances in Applied Mathematics
... approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment ...
Tópico(s): Polynomial and algebraic computation
2010 - SIGSAM | ACM communications in computer algebra
Tewodros Amdeberhan, Olivier Espinosa, V. H. Moll, Armin Straub,
... mathematics from Temple University under the supervision of Doron Zeilberger. His main interest lies in WZ-theory, number ...
Tópico(s): Advanced Mathematical Theories and Applications
2010 - Taylor & Francis | American Mathematical Monthly
Manuel Kauers, Christoph Koutschan, Doron Zeilberger,
We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .
Tópico(s): Polynomial and algebraic computation
2009 - National Academy of Sciences | Proceedings of the National Academy of Sciences
Many explicit determinant evaluations can be automatically conjectured, and then rigorously automatically proved, once we suspect that they belong to the Holonomic Ansatz.
Tópico(s): Algebraic structures and combinatorial models
2007 - Birkhäuser | Annals of Combinatorics
Arvind Ayyer, Doron Zeilberger,
We show that the generating function (in $n$) for the number of walks on the square lattice with steps $(1,1), (1,-1), (2,2)$ and $(2,-2)$ from $(0,0)$ to $(2n,0)$ in the region $0 \leq y \leq w$ satisfies a very special fifth order nonlinear recurrence relation in $w$ that implies both its numerator and denominator satisfy a linear recurrence relation.
Tópico(s): Algorithms and Data Compression
2007 - Electronic Journal of Combinatorics | The Electronic Journal of Combinatorics
Doron Zeilberger, David M. Bressoud,
Let (y)a=(1-y)(1-qy)⋯(1-qa-1y). We prove that the constant term of the Laurent polynomial ∏1⩽i<j⩽n(xi/xj)ai(qxj/xi)aj, where x1,…,xn,q are commuting indeterminates and a1,…,an are non-negative integers, equals (q)a1+⋯+an/(q)a1…(q)an. This settles in the affirmative a conjecture of George Andrews (in: R.A. Askey, ed., Theory and Applications of Special Functions, Academic Press, New York, 1975, 191–224].
Tópico(s): Analytic Number Theory Research
2006 - Elsevier BV | Discrete Mathematics
Stavros Garoufalidis, Thang T. Q. Lê, Doron Zeilberger,
We state and prove a quantum generalization of MacMahon's celebrated Master Theorem and relate it to a quantum generalization of the boson-fermion correspondence of physics.
Tópico(s): Advanced Operator Algebra Research
2006 - National Academy of Sciences | Proceedings of the National Academy of Sciences
Andrew V. Sills, Doron Zeilberger,
We present a case study in experimental yet rigorous mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that automatically conjectures, and then automatically proves, closed-form expressions extending Dyson's celebrated constant-term conjecture.
Tópico(s): Advanced Mathematical Identities
2006 - Taylor & Francis | Experimental Mathematics
Tópico(s): Topological and Geometric Data Analysis
2004 - Birkhäuser | Annals of Combinatorics
The N-heap Wythoff’s game is a two-player impartial game with N piles of tokens of sizes $$ A^1, \ldots , A^N, A^1 \leq \ldots \leq A^N $$ Players take turns removing any number of tokens from a single pile, or removing (a1,..., a N ) from all piles - ai tokens from the i-th pile, providing that $$ 0 \leq a_i \leq A^i, \sum^{N}_{i=1} a-i > = \quad \mathrm{and} \quad a_1 \bigotimes \ldots \bigotimes a_N = 0 $$ where ⊕ is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) ...
Tópico(s): Sports Analytics and Performance
2004 - Birkhäuser | Annals of Combinatorics
In this article, dedicated with admiration and friendship to chaos and difference (and hence recurrence) equations guru Saber Elaydi, I give a new approach and a new algorithm for Chomp, David Gale's celebrated combinatorial game. This work is inspired by Xinyu Sun's "ultimate-periodicity" conjecture and by its brilliant proof by high-school student Steven Byrnes. The algorithm is implemented in a Maple package BYRNES accompanying this article. By looking at the output, and inspired by previous work ...
Tópico(s): Artificial Intelligence in Games
2004 - Taylor & Francis | The Journal of Difference Equations and Applications
Dominique Foata, Doron Zeilberger,
Tópico(s): Stochastic processes and statistical mechanics
2003 - Birkhäuser | Algebra Universalis
Aaron Robertson, Dan Saracino, Doron Zeilberger,
Define $ S_{k}^n (\alpha) $ to be the set of permutations of {1, 2,...,n} with exactly k fixed points which avoid the pattern $ \alpha\in S_m $ . Let $ S_{k}^n (\alpha) $ be the size of $ S_{k}^n (\alpha) $ . We investigate $ S_{n}^0 (\alpha) $ for all $ \alpha\in S_3 $ as well as show that $ s_{n}^{k} (132) = s_{n}^{k}(213) = s_{n}^{k}(321)\quad\mathrm{and}\quad s_{n}^{k}(231) = s_{n}^{k}(312)\quad\mathrm{for\quad all}\quad 0\leq k\leq n $ .
Tópico(s): Algorithms and Data Compression
2002 - Birkhäuser | Annals of Combinatorics
Tewodros Amdeberhan, Doron Zeilberger,
Using a recurrence derived from Dodgson's Condensation Method, we provide numerous explicit evaluations of determinants. They were all conjectured, and then rigorously proved, by computer-assisted methods, that should be amenable to full automation. We also mention a first step towards that goal, our Maple package, DODGSON, that automates the special case of Hankel and Toeplitz hypergeometric determinants.
Tópico(s): SAS software applications and methods
2001 - Elsevier BV | Advances in Applied Mathematics
A “meta” (pseudo-) algorithm is described that, for any fixed k, finds a fast (O(log(a))) algorithm for playing 3-rowed Chomp, starting with the first, second, and third rows of lengths a, b, and c, respectively, where c ≤ k, but a and b are arbitrary.
Tópico(s): Sports Analytics and Performance
2001 - Elsevier BV | Advances in Applied Mathematics
Dominique Foata, Doron Zeilberger,
Babson and Steingrı́msson have recently introduced seven new permutation statistics, that they conjectured were all Mahonian (i.e., equi-distributed with the number of inversions). We prove their conjecture for the first four and also prove that the first and the fourth are even Euler–Mahonian. We use two different, in fact, opposite, techniques. For three of them we give a computer-generated proof, using the Maple package ROTA, that implements the second author's “Umbral Transfer Matrix Method.” ...
Tópico(s): semigroups and automata theory
2001 - Elsevier BV | Advances in Applied Mathematics
Tópico(s): Mathematics and Applications
2000 - Elsevier BV | Journal of Combinatorial Theory Series A
Clara S. Chan, David P. Robbins, David S. Yuen,
... Editor's note: After this paper was circulated, Doron Zeilberger [1998] proved Conjecture 1, using the authors' reduction ...
Tópico(s): Coding theory and cryptography
2000 - Taylor & Francis | Experimental Mathematics