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...
In a wonderful essay[W] on Experimental Mathematics, Herb Wilf outlines the four steps of doing Experimental Mathematics, in the way it is usually practiced today. 1. Wondering, by a human, what a...
Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
Toward a Crlmbinatorial Proof of the Jacobian Conjecture? (2009)
Dominique Foata taught us how to do algebra and special functions com-binatorially. Now ~ndr; Joyal and his diciples teach us how to do calculus combinatorially. The first part of this paper will...
Theorem ( q-extension of (4.2) in [F2]): (2009)
As always in q-theory, (X; Q)n will stand for the product (1 − X)(1 − QX)...(1 − Q n−1 X), and when the ”base ” Q is q, we will abbreviate (X; q)n to (X)n. For any Laurent polynomial f in...
PROOF OF IRA GESSEL’S LATTICE PATH CONJECTURE (2009)
Manuel Kauers, Doron Zeilberger
Abstract. 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 ≥ 0, y ≥...
Manuel Kauers, Christoph Koutschan, Doron Zeilberger
In the historic conference Combinatoire Énumérative [6] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre...
TOWARDS A WZ EVOLUTION OF THE MEHTA INTEGRAL* (2009)
Abstract. The celebrated Mehta integral is shown to be equivalent to a simple algebraicdifferential identity, which is completely routine for any fixed number of variables. Key words. WZ form,...
This is the fifth, and last, installment of the saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. Here we extend the powerful Goulden-Jackson...
The transfer-matrix method, like the Principle of Inclusion-Exclusion and the Möbius (2009)
inversion formula, has simple theoretical underpinnings but a very wide range of applicability.
PROOF OF A CONJECTURE OF CHAN, ROBBINS, AND YUEN ∗ (2009)
Abstract. Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain n(n −...
Deconstructing the Zeilberger algorithm † (2009)
By looking under the hood of Zeilberger’s algorithm, as simplified by Mohammed and Zeilberger, it is shown that all the classical hypergeometric closed-form evaluations can be discovered ab initio,...
Dominique Foata, Doron Zeilberger
Abstract. Further computations are made on the traditional coupon collector’s problem when the collector shares his harvest with his younger brothers. When the book of the p-th brother of the...
Determinants through the looking glass (2009)
Tewodros Amdeberhan, Doron Zeilberger
dedicated to dominique foata on his 65th birthday Using a recurrence derived from Dodgson’s Condensation Method, we provide numerous explicit evaluations of determinants. They were all conjectured,...
Babson-Steingrímsson statistics are indeed Mahonian (and sometimes even Euler-Mahonian (2009)
Dominique Foata, Doron Zeilberger
Babson and Steingrímsson 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...
In How Many Ways Can You Reassemble Several Russian Dolls? (2009)
When my brilliant student Thotsaporn "Aek" Thanatipanonda asked me about how many ways can one cover n identical twins, it rang a Bell back to 1981 (see the article). As usual, the method of teaching...
A Multi-Set Identity for Partitions (2009)
Regev, Amitai, Zeilberger, Doron
We prove that the multiset {(RightArmLength,LeftArmLength)} ranging over all cells of all Ferrers diagrams with n cells equals the multiset {(RightArmLength,LegLength)} ranging over all cells of all...
The Mahonian probability distribution on words is asymptotically normal (2009)
Canfield, E. Rodney, Janson, Svante, Zeilberger, Doron
The Mahonian statistic is the number of inversions in a permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function for this statistic is the $q$ analog of the...
Some Nice Sums are Almost as Nice if you turn them Upside Down (2009)
Apagodu, Moa, Zeilberger, Doron
We represent the sums $\sum_{k=0}^{n-1}{n \choose k}^{-2}$, $\sum_{k=0}^m{m\choose k}^{-1}{a\choose n-k}^{-1}$, $\sum_{k=0}^{n-1}\frac{q^{-k(k-1)}}{{\genfrac{[}{]}{0pt}{}{n}{k}}_q}$, and the sum of...
Finite Analogs of Szemer\'edi's Theorem (2009)
One of the "deepest" theorems in mathematics is Endre Szemer\'edi's theorem about the inevitability of arithmetical progressions. Here we try to nibble at it, by doing "finite" analogs. This is...
An Experimental Mathematics Perspective on the Old, and still Open, Question of When To Stop? (2009)
Medina, Luis A., Zeilberger, Doron
In a recent article in American Scientist, Theodore Hill described a coin-tossing game whose pay-off is the number of heads over the total number of throws. Suppose that at a given point during the...
A Symbolic Computational Approach to a Problem Involving Multivariate Poisson Distributions (2009)
Sontag, Eduardo, Zeilberger, Doron
Multivariate Poisson random variables subject to linear integer constraints arise in several application areas, such as queuing and biomolecular networks. This note shows how to compute conditional...
Paul Erdos claimed that mathematics is not yet ready to settle the 3x+1 conjecture. I agree, but very soon it will be! With the exponential growth of computer-generated mathematics, we (or rather our...
On the number of walks on a regular Cayley tree (2009)
Rowland, Eric, Zeilberger, Doron
We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for...
Two Dimensional Directed Lattice Walks with Boundaries (2008)
Arvind Ayyer, Doron Zeilberger
We present general algorithms (fully implemented in Maple) for calculations of various quantities related to constrained directed walks for a general set of steps on the square lattice in two...
The collector's brotherhood problem using the Newman-Shepp symbolic method (2008)
Dominique Foata, Doron Zeilberger
Abstract. Further computations are made on the traditional coupon collector’s problem when the collector shares his harvest with his younger brothers. When the book of the p-th brother of the...
Babson-Steingrímsson statistics are indeed Mahonian (and sometimes even Euler-Mahonian (2008)
Dominique Foata, Doron Zeilberger
Abstract: Babson and Steingrímsson have recently introduced seven new permutation statistics, that they conjectured were all Mahonian (i.e. equi-distributed with the number of inversions). We prove...
Abstract: This is the fifth, and last, installment of the saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. Here we extend the powerful...
Proof of a Conjecture of Amitai Regev about Three-Rowed Young Tableaux (and much more!) (2008)
Shalosh B. Ekhad, Doron Zeilberger
Consider lattice paths in the three-dimensional cubic lattice, with unit positive steps, that always stay in the region x ≥ y ≥ z. Let Ab1,b2,b3(n) be the number of such walks of length n−(b1...
Doron Zeilberger, Thomas Kuhn, Science His
be thus dubbed the Paradigm Paradigm. Doron Zeilberger, not-yet so famously, believes that Mathematics, in the future, will be ansatz-based, so my approach to mathematical research could be called...
An Umbral Approach to the Hankel Transform for Sequences (2008)
Recall that the binomial transform of a sequence {an} ∞ n=0, let’s call it BIN(a) = {bn} ∞ n=0, is defined by bn = n� k=0 n ak,
5. Exploiting Symmetry and Algebraic Relations (2008)
Andrew V. Sills, Doron Zeilberger, Dedicated Freeman, John Dyson, Irving John Good
Discrete Calculus Problem: Find the minimal value of (2008)
Aaron Robertson, Doron Zeilberger
Abstract: We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1, n] can have is n2 22 + O(n). This revised version fills in a minor and subtle gap...
Doron Zeilberger, Greg Warrington
made a seemingly amazing conjecture. Let a and b be relatively prime positive integers and let n be a positive integer. There are exactly �n lattice paths from (0, 0) to (nab, nab), with...
In 1921, George Pólya[P] (see also [DS], ch. 7) famously proved that if a drunkard leaves a pub situated at the origin, in a d-dimensional Manhattan (w/o Broadway), and where he is allowed to only...
I Have a Dream Automatic CounTilings (2008)
One day it would be possible to write in English, or in an English-like super-high-level programming language, the following command: Write a Maple program that inputs an arbitrary positive integer...
EXPERIMENTS WITH A POSITIVITY PRESERVING OPERATOR (2008)
Manuel Kauers, Doron Zeilberger
Abstract. We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves...
Tewodros Amdeberhan, Doron Zeilberger, Xi Yj Axiyj
Abstract. We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of...
A short proof is not as satisfying as a long one. A good mathematical proof should do much more than just prove the theorem. It should amuse, instruct, and satisfy our deeper love of knowledge, which...
A bijectional attack on the Razumov-Stroganov conjecture (2008)
Ayyer, Arvind, Zeilberger, Doron
We attempt to prove the Razumov-Stroganov conjecture using a bijectional approach. We have been unsuccessful but we believe the techniques we present can be used to prove the conjecture.
Dedicated to the memory of my grandfather Paul Alexander (2008)
Doron Zeilberger, Dr. Paul Alex, Dr. Phil Rer, Nat Chemie
This is my first visit to Leipzig. My main reasons for coming here are personal: to look up the graves and dwellings of my great-grandparents, Salomon and Rebecka Alexander, and to explore the city...
mn = 1, − 1 where � ′ means that every term only occurs once (for example 1/15 = 1/(16 − 1) is only added once even though 16 = 4 2 = 2 4). Proof: Let R denote the set of all integers larger...
Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
For example, A(5 + z −1 Let 1 + z3 1z −2 (2008)
Doron Zeilberger, The Anti-symmetrizer
is defined by: A(P)(z1,..., zn) = � (−1) inv(π) P (zπ(1),..., zπ(n)), π∈Sn where Sn is the group of permutations on {1,..., n} and inv(π) is the number of inversions of π (the number of...
for all n ≥ 1. Two conjectures were proposed on the game by Fraenkel [7]. (2008)
Abstract. The N-heap Wythoff’s game is a two-player impartial game with N piles of tokens of sizes A1,..., AN, A1 ≤ · · · ≤ AN. Players take turns removing any number of tokens from a single...
An enumerative retailer studies one problem at a time. On the other hand, an enumerative wholesaler studies a family of problems, and tries to design algorithms that can be implemented to solve lots...
Dedicated to Two Combinatorial Giants: Gian-Carlo Rota (April 27, 1932- April 18, (2008)
Shalosh B. Ekhad, Doron Zeilberger, Rd Birthdays
respectively.
RANDOM WALK IN A WEYL CHAMBER 1 (2008)
Ira M. Gessel, Doron Zeilberger
Abstract: The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge x1 ≥ x2 ≥... ≥ xn (which is a Weyl chamber for the symmetric...
Kauers, Manuel, Koutschan, Christoph, Zeilberger, Doron
In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre...
Proof of Ira Gessel's Lattice Path Conjecture (2008)
Kauers, Manuel, Koutschan, Christoph, Zeilberger, Doron
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...
The Quasi-Holonomic Ansatz and Restricted Lattice Walks (2008)
Kauers, Manuel, Zeilberger, Doron
The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example,...
Moa Apagodu, Doron Zeilberger, Dedicated Sergei, Abramov Five-dozen
In the old days, when one had to find some sequence, a(n), there were two extremes. In the lucky case, one had an explicit formula. For example, the probability of tossing a fair coin 2n times and...
Borsuk Conjecture and Kahn-Kalai counterexample ([Thom, Cor 1.3.2], [KK]), (2008)
Philip Matchett Wood, Committee Profs, Van Vu, József Beck, Endre Szemerédi, Doron Zeilberger
Fisher’s and generalized Fisher’s inequalities ([Thom, Lem 1.2.1], [vLW, p. 222]),
Abstract. This is the third part of the five-part saga on the umbral transfermatrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. In this article we describe the Maple package ZOO...
Comments: The great enumerator Germain Kreweras empirically discovered this intriguing fact, (2008)
Manuel Kauers, Doron Zeilberger, For Example, Found Other
and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators,
Abstract: The Berger-Felzenbaum-Fraenkel approach to Covering Systems is exposited. In particular their gorgeous proof of the famous an = an−1 theorem for exact covering systems (found...
Searching for Strange Hypergeometric Identities By Sheer Brute Force (2008)
Apagodu, Moa, Zeilberger, Doron
We describe a systematic search for all strange hypergeometric identities up to a certain complexity with sheer brute force that lead us to the discovery of two new infinite families of closed-form...
Efficient Counting and Asymptotics of $k$-noncrossing tangled-diagrams (2008)
Chen, William Y. C., Qin, Jing, Reidys, Christian M., Zeilberger, Doron
In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a...
Comments: The great enumerator Germain Kreweras empirically discovered this intriguing fact, (2008)
Manuel Kauers, Doron Zeilberger, For Example, Found Other
and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators,
EFFICIENT COUNTING AND ASYMPTOTICS OF k-NONCROSSING TANGLED-DIAGRAMS (2008)
Jing Qin, Christian M. Reidys, Doron Zeilberger, Jing Qin, ...
Abstract. In this paper we enumerate k-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are 1,..., n have degree ≤ 2, and are arranged in increasing order in a...
ABSTRACT OF THE DISSERTATION Enumeration Schemes for Pattern-Avoiding Words and Permutations (2008)
Kristin Pudwell, Doron Zeilberger, Lara Kristin Pudwell, Dissertation Director, Doron Zeilberger
Let p = p1 · · · pn ∈ Sn and q = q1 · · · qm ∈ Sm. We say that p contains q as a pattern if there are indices 1 ≤ i1 < · · · < im ≤ n such that pij < pik ⇐ ⇒ qi < qk;...
A bijectional attack on the Razumov-Stroganov conjecture (2008)
Arvind Ayyer, Doron Zeilberger
We attempt to prove the Razumov-Stroganov conjecture using a bijectional approach. We have been unsuccessful but we believe the techniques we present can be used to prove the conjecture. 1
Yuri Bahturin, Doron Zeilberger
Abstract: We consider a certain decomposition of the matrix algebra Mn(F), where F is a field. The commutation relations of that decomposition yield an n 2 × n 2 matrix M Mn(F), which determines the...
Dominique Foata, Doron Zeilberger, To Marco Schutzenberger, In Memoriam
classic proof of a recurrence
Combinatorial Proofs Of Capelli's And Turnbull's Identities From Classical Invariant Theory (2007)
Dominique Foata, Doron Zeilberger
this paper, we give short combinatorial proofs of Capelli's and Turnbull's identities, and raise the hope that someone else will use our approach to prove the new Howe-Umeda-Kostant-Sahi...
A High-School Algebra, Wallet-Sized Proof, of the Bieberbach Conjecture [After L. Weinstein] (2007)
Shalosh B. Ekhad, Doron Zeilberger
This paper makes explicit the power of formal math, that is (too) subtly hidden in Weinstein's original presentation.
A Wz Proof Of Ramanujan's Formula For (2007)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger
0.37> \Gamma(3=2 + n) \Gamma(3=2)\Gamma(n + 1) = 1 X k=0 (\Gamma1) k (4k + 1) (1=2) 2 k (\Gamman) k k! 2 (3=2 + n) k : (3) To prove it for all positive integers n, we call the summand divided by...
There Are MORE THAN 2**(n/17) n-LETTER TERNARY SQUARE-FREE WORDS (2007)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger, Improving On Brinkhuis
: We prove that the `connective constant' for ternary square-free words is at least 2 1=17 = 1:0416 : : :, improving on Brinkhuis and Brandenburg's lower bounds of 2 1=24 = 1:0293 : : : and...
Combinatorial Proofs Of Capelli's And Turnbull's Identities From Classical Invariant Theory (2007)
Dominique Foata, Doron Zeilberger
this paper, we give short combinatorial proofs of Capelli's and Turnbull's identities, and raise the hope that someone else will use our approach to prove the new Howe-Umeda-Kostant-Sahi...
Can Have, Aaron Robertson, Doron Zeilberger
: We prove the statement of the title, thereby solving a $100 problem of Ron Graham. This was solved independently by Tomasz Schoen. Tianjin, June 29, 1996: In a fascinating invited talk at the SOCA...
Amitai Regev, Doron Zeilberger
The notion of hook number was originally defined for the cells of the Young diagram D(a): = f(i; j) j 1 j k; 1 i a k\Gammaj+1 g of a partition a = (a 1; : : : ; a k). The hook number, h(x), of a cell...
Abstract: H.N.V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple). Preface I have never met Henry Gould in...
Like all mathematicians who grew up in Israel, I was always in awe of the members of the faculty of the Hebrew University at Jerusalem, roughly the Israeli analog of Princeton. Hence I felt a huge...
Last spring, Jon Borwein and David Bradley[BB] came up with an intriguing conjecture. They stated several equivalent statements, one of them, due to Wenchang Chu, states: Superficially, this looks...
itre, Marco Schutzenberger, pursued so vigorously and successfully by the Ecole (2007)
Albert Einstein used to be amazed how eective mathematics is in science. Myself, I am amazed how eective combinatorics is in mathematics. Especially fruitful is the `formal language approach' of...
Doron Zeilberger, Last Spring, Jon Borwein
stated several equivalent statements, one of them, due to Wenchang Chu, states: n
Research Announcement: The Transcendence of e + and e (2007)
There are @ real numbers, while there are only @ 0 algebraic numbers. Hence the probability that a randomly chosen real number is algebraic equals
Doron Zeilberger, Paul Halmos, Steven Krantz
Abstract: A Powerful new combinatorial theory, superseding, and sometimes trivializing, mathematical analysis, is introduced. It is illustrated by an exact determination of Bloch's constant, a...
A Heterosexual Mehler Formula for the Straight Hermite Polynomials ( (2007)
Abstract: The celebrated Foata combinatorial model for Hermite polynomials, and his seminal and beautiful proof of the Mehler formula, are straightened to deal with two sexes rather than one, with...
A New Proof that there are 2 n Possible Outcomes on Tossing a Coin n Times (2007)
A short proof is not as satisfying as a long one. A good mathematical proof should do much more than just prove the theorem. It should amuse, instruct, and satisfy our deeper love of knowledge, which...
MR Subject Classications: 05A (2007)
To think in a computerized way is an important matter: : : to go with it to the end of the limit of possibilities, and there to develop new, unpredictable ones. | David
The Umbral Transfer-Matrix Method. IV. Counting SelfAvoiding Polygons and Walks, this article (2007)
To think in a computerized way is an important matter... to go with it to the end of the limit of possibilities, and there to develop new, unpredictable ones. — David Avidan (Free translation of an...
Doron Zeilberger, Francesco Brenti's Conjecture
In the very rst day of the Oberwolfach meeting on Enumerative Combinatorics and the Symmetric Group (Jan. 16-20, 1995), during the morning session, Francesco Brenti gave a wonderful talk about his...
Like all mathematicians who grew up in Israel, I was always in awe of the members of the faculty of the Hebrew University at Jerusalem, roughly the Israeli analog of Princeton. Hence I felt a huge...
The Umbral Transfer-Matrix Method: II. Counting Plane Partitions (2007)
Abstract: We continue the 5-part saga on the Umbral Transfer-Matrix Method, based on GianCarlo Rota's seminal notion of the umbra. In this article we describe the Maple package PPar that...
Doron Zeilberger, David Bressoud, Gaurav Bhatnagar, Anders Bjorner, Jonathan Borwein, Francesco Brenti, ...
Two stones build two houses. Three build six houses. Four build four and twenty houses. Five build hundred and twenty houses. Six build Seven hundreds and twenty houses. Seven build five thousands...
Thanatipanonda, Thotsaporn ``Aek'', Zeilberger, Doron
We develop a finite-state automata approach, implemented in a Maple package {\tt ToadsAndFrogs} available from our websites, for conjecturing, and then rigorously proving, values for large families...
Experiments with a Positivity Preserving Operator (2007)
Kauers, Manuel, Zeilberger, Doron
We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of...
Two Dimensional Directed Lattice Walks with Boundaries (2007)
Ayyer, Arvind, Zeilberger, Doron
We present general algorithms (fully implemented in Maple) for calculations of various quantities related to constrained directed walks for a general set of steps on the square lattice in two...
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 ≤ y ≤ w...
TrivializingGeneralizations of some Izergin-Korepin-type Determinants (2007)
Tewodros Amdeberhan, Doron Zeilberger
We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our...
Ayyer, Arvind, Zeilberger, Doron
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
The Quantum MacMahon Master Theorem (2006)
Stavros Garoufalidis, Thang Tq Lê, Doron Zeilberger
Abstract. 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. 1.
Advances in Applied Mathematics 37 (2006) 139–152 (2006)
Multi-variable Zeilberger, Moa Apagodu, Doron Zeilberger
www.elsevier.com/locate/yaama
A Proof of the Loehr-Warrington Amazing TEN to the Power n Conjecture (2005)
Ekhad, Shalosh B., Vatter, Vince, Zeilberger, Doron
We prove, via 30 seconds of Maple computation, that there are 10^n words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be...
Pythagorean primes and palindromic continued fractions, Integers 5 (2005)
Arthur T. Benjamin, Doron Zeilberger
In this note, we prove that every prime of the form 4m+1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued...
Andrew V. Sills, Doron Zeilberger, Dedicated Freeman, John Dyson, Irving John Good
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...
Andrew V. Sills, Doron Zeilberger, Dedicated Freeman, John Dyson, Irving John Good
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...
Mohamud Mohammed, Doron Zeilberger, Logisch-philosophische Abhandlung
www.elsevier.com/locate/jsc Sharp upper bounds for the orders of the recurrences output by the Zeilberger and q-Zeilberger algorithms ✩
AMS Subject Classification: 05A05 Abstract. The old workhorse called linearity of expectation, by which it is often very easy to compute the expectation (alias first moment) of interesting...
The quantum MacMahon Master Theorem (2003)
Garoufalidis, Stavros, Le, Thang TQ., Zeilberger, Doron
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.
Annals of Combinatorics On Fraenkel’s N-Heap Wythoff’s Conjectures (2003)
Abstract. The N-heap Wythoff’s game is a two-player impartial game with N piles of tokens of sizes A1,..., AN, A1 ≤ ·· · ≤ AN. Players take turns removing any number of tokens from a single...
Chomp, Recurrences and Chaos(?) (2003)
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...
Refined Restricted Permutations (2002)
Robertson, Aaron, Saracino, Dan, Zeilberger, Doron
Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We...
www.elsevier.com/locate/yaama Computerized deconstruction ✩ (2002)
The inequality (DERRIDA + TURING)>(DERRIDA) + (TURING) will be illustrated by computerized deconstruction of Roger Apéry’s miraculous proofs of irrationality.
Communicated by the Managing Editors (2000)
dedicated to the memory of gian-carlo rota We lay the foundations for the Umbral Transfer-Matrix Method, based on Gian-Carlo Rota's seminal notion of the ``umbra' ' as a linear...
on The Goulden–Jackson Cluster Method for Cyclic Words (2000)
Anne E. Edlin, Doron Zeilberger
The powerful Goulden–Jackson Cluster Method, which generates generating functions enumerating words that avoid, as factors, a prescribed finite set of “mistakes,” is adapted to handle cyclic...
Robertson, Aaron, Wilf, Herb, Zeilberger, Doron
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
The Number of Permutations With A Prescribed Number of 132 and 123 Patterns (1999)
Ekhad, Shalosh B., Robertson, Aaron, Zeilberger, Doron
Here we present the reasoning behind, and program to find, the generating functions for the number of permutations in the title. The article duals as the "accompanying" Maple package.
A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs (1999)
Dominique Foata, Doron Zeilberger
This paper is dedicated to Gian-Carlo Rota, on his millionth2’s birthday. Abstract. We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a...
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Proof of a Conjecture of Chan, Robbins, and Yuen (1998)
Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain $n(n-1)/2$-dimensional...
The impact of the computer on present and especially future mathematics is illustrated by means of the iconic example of WZ theory.
There are More Than 2**(n/17) n-Letter Ternary Square-Free Words (1998)
We prove that the `connective constant' for ternary square-free words is at least $2^{1/17} = 1.0416 ... $, improving on Brinkhuis and Brandenburg's lower bounds of $2^{1/24}=1.0293 ...$ and...
The Combinatorial Astrology of Rabbi Abraham Ibn Ezra (1998)
Rabbi Abraham Ben Meir Ibn Ezra's(1089-1164) computation of the seventh row of the so-called Pascal-Chu triangle is described.
The Abstract Lace Expansion (1998)
David Brydges and Thomas Spencer's Lace Expansion is abstracted, and it is shown how it sometimes gives rise to sieves.
Proof of Conway's Lost Cosmological Theorem (1998)
Ekhad, Shalosh B., Zeilberger, Doron
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for which no extant proof existed, is given a computer-generated proof, hopefully for good.
A Comparison Of Two Methods For Random Labelling of Balls by Vectors of Integers (1998)
A combinatorial problem that came up in Combinatorial Chemistry is solved.
Dodgson's Determinant-Evaluation Rule proved by Two-Timing Men and Women (1998)
The Rev. Dodgson's determinant condensation rule is given a bijective proof.
The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns (1998)
Noonan, John, Zeilberger, Doron
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always...
A Heterosexual Mehler Formula for the Straight Hermite Polynomials (A La Foata) (1998)
The celebrated Foata combinatorial model for Hermite polynomials, and his seminal and beautiful proof of the Mehler formula, are straightened to deal with two sexes rather than one, with the...
The Goulden-Jackson Cluster Method: Extensions, Applications and Implementations (1998)
Noonan, John, Zeilberger, Doron
The powerful (and so far under-utilized) Goulden-Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of `dirty...
A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs (1998)
Foata, Dominique, Zeilberger, Doron
We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon...
Curing the Andrews syndrom (1998)
Ekhad, Shalosh B., Zeilberger, Doron
George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric sums is...
A binomial coefficient identity associated to a conjecture of Beukers (1998)
Ahlgren, Scott, Ekhad, Shalosh B., Ono, Ken, Zeilberger, Doron
A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.
How Much Should a 19th-Century French Bastard Inherit (1998)
Catalan's formula, for the portion of the inheritance that a legitimate child of a 19th-century deceased French gentleman should receive, is given a new proof (using Difference Operators), and...
Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten (1998)
Krattenthaler, Christian, Zeilberger, Doron
A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math\. {\bf 4} (1995), 87--96) that arose in the study of...
Plane Geometry: An Elementary School Textbook (ca. 2050) (1998)
Ekhad, Shalosh B., Zeilberger, Doron
In this reverse archeological find, we see how mathematics will be written in fifty years.
Aufgabe VII.47 of Polya-Szego Implies Robbins's Multi-Integral Evaluation (1998)
It is remarked that Dave Robbins's Expression for the integral of an alternant over the unit simplex (math.CO/9805108) follows immediately from an excercise in Polya-Szego.
Enumeration Schemes And (More Importantly) Their Automatic Generation (1998)
It is way too soon to teach our computers how to become full-fledged humans. It is even premature to teach them how to become mathematicians, it is even unwise, at present, to teach them how to...
A Classic Proof of a Recurrence for a Very Classical Sequence (1998)
Foata, Dominique, Zeilberger, Doron
By practicing the philosophy of our beloved late master, Marco Schutzenberger, to whose memory this article is dedicated, we give an insightful bijective proof of the three-term recurrence satisfied...
Hypergeometric Series Acceleration Via the WZ method (1998)
Amdeberhan, Tewodros, Zeilberger, Doron
Based on the WZ method, some series acceleration formulas are given. These formulas allow to write down an infinite family of parametrized identities from any given identity of WZ type. Further, this...
q-Apery Irrationality Proofs by q-WZ Pairs (1998)
Amdeberhan, Tewodros, Zeilberger, Doron
Using WZ forms, Apery-style proofs of the irrationality of the q-analogues of the Harmonic seires and Ln(2) are given. For the q-analogue of Ln(2), this method of proof produces an improved...
A 2-coloring of [1,n] can have (n^2)/22 + O(n) monochromatic Schur triples, but not less! (1998)
Robertson, Aaron, Zeilberger, Doron
We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1,n] can have is (n^2)/22 + O(n). This was solved independently by Tomasz Schoen.
Automated counting of LEGO towers (1998)
H. N. V. Temperley's method for counting vertically convex polyominoes is modified, generalized, and most importantly, programmed (in Maple).
A heterosexual Mehler formula for the straight Hermite polynomials (1998)
Abstract: The celebrated Foata combinatorial model for Hermite polynomials, and his seminal and beautiful proof of the Mehler formula, are straightened to deal with two sexes rather than one, with...
A Pentagonal Number Sieve (1998)
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger
We prove a general "pentagonal sieve" theorem that has corollaries such as the following. First, the number of pairs of partitions of n that have no parts in common is p(n) 2 \Gamma p(n...
CURING The ANDREWS SYNDROME (1998)
Shalosh Ekhad And, Shalosh B. Ekhad, Doron Zeilberger
: George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric...
A 2-Coloring Of [1,N] Can Have (1/22)N² +O(N) MONOCHROMATIC SCHUR TRIPLES, BUT NOT LESS! (1998)
Aaron Robertson, Doron Zeilberger
: We prove the statement of the title, thereby solving a $100 problem of Ron Graham. This was solved independently by Tomasz Schoen. Tianjin, June 29, 1996: In a fascinating invited talk at the SOCA...
AMS Subject Classification: 05A15 Abstract. The notion of Enumeration Scheme is introduced and applied to the problem of counting permutations with forbidden patterns. Most importantly, the process...
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
[50] Develop computer programs for simplifying sums
Dodgson's determinant-evaluation rule proved by TWO-TIMING (1997)
Doron Zeilberger, I Reverend, Charles Lutwidge
det
Hypergeometric series acceleration via the WZ method (1997)
Tewodros Amdeberhan, Doron Zeilberger, Typeset Ams-tex
Dedicated to Herb Wilf on his one million-first birthday Abstract. Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an infinite family of...
Proof of Conway's Lost Cosmological Theorem (1997)
Shalosh B. Ekhad, Doron Zeilberger
th(B i+1 )=length(B i ) ! . This is an immediate consequence of The Cosmological Theorem: There exists an integer N such that every string decays in at most N days to a compound of common and...
How to do MONTHLY problems with your computer (1997)
Istvan Nemes, Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
this article (PWZ) have just written a book [8] that describes the theoretical foundations of the solution of this problem, and also gives the software by means of which everyone can perform these...
Marko Petkovsek, Donald E. Knuth, Doron Zeilberger, Herbert Wilf
Contents Foreword vii AQuickStart... ix I Background 1 1 Proof Machines 3 1.1 Evolutionoftheprovinceofhumanthought .............. 3 1.2 Canonicalandnormalforms....................... 7 1.3...
How to do MONTHLY problems with your computer (1997)
István Nemes, Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
this article (PWZ) have just written a book [8] that describes the theoretical foundations of the solution of this problem, and also gives the software by means of which everyone can perform these...
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
This page intentionally left blank [50] Develop computer programs for simplifying sums
Donald E. Knuth, Doron Zeilberger, Herbert S. Wilf
This page intentionally left blank [50] Develop computer programs for simplifying sums
Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, Communicated George Andrews
We prove a general ``pentagonal sieve' ' theorem that has corollaries such as the following. First, the number of pairs of partitions of n that have no parts in common is p(n) 2 &...
ARTICLE NO. AM970565 q-Apery ´ Irrationality Proofs by q-WZ Pairs (1997)
Tewodros Amdeberhan, Doron Zeilberger
Using WZ pairs, Apery-style ´ proofs of the irrationality of the q-analogues of the Harmonic series and LnŽ. 2 are given. For the q-analogue of LnŽ. 2, this method produces an improved...
Dominique Foata, Doron Zeilberger, Communicated George Andrews
to marco schutzenberger, in memoriam Richard Stanley [St96] has recently narrated the fascinating story of how the classical Schroder [Sch1870] numbers s(n) are even more classical than was...
(Communicated by Ronald Graham) (1997)
Shalosh B. Ekhad, Doron Zeilberger
existed, is given a new proof, this time hopefully for good. One of the most intriguing sequences [CG], [F], [SP], [V] is Conway’s [C] 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,.... It...
Proof of the Refined Alternating Sign Matrix Conjecture (1996)
Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills,...
The enumeration of permutations with a prescribed number of “forbidden” patterns (1996)
Abstract. We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden ' patterns, that seems to indicate that the enumerating...
The Enumeration Of Permutations With A Prescribed Number Of "Forbidden" Patterns (1996)
. We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is...
Hypergeometric Series Acceleration via the WZ Method (1996)
Tewodros Amdeberhan, Doron Zeilberger
. Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an infinite family of parametrized identities from any given identity of WZ type. Further,...
Proof of the Refined Alternating Sign Matrix Conjecture (1996)
. Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n) := 1!4!7! \Delta \Delta \Delta (3n \Gamma 2)! n!(n + 1)! \Delta...
Proof of the refined alternating sign matrix conjecture (1996)
Doron Zeilberger, Abstract Mills, Rumsey Conjectured, Zeilberger Proved
the number of alternating sign matrices of order n equals
Proof of the refined alternating sign matrix conjecture (1996)
Abstract. Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order n equals A(n):= 1!4!7! ···(3n − 2)! n!(n +1)!···(2n − 1)!....
FOR ERDOS ¨ PAL, ´ IN MEMORIAM (1996)
David Brydges and Thomas Spencer’s lace expansion is abstracted and it is shown how it sometimes gives rise to sieves. � 1997 Academic Press LACES DEFINITION. Let P be a finite set of properties....
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of ‘‘forbidden’ ’ patterns that seems to indicate that the enumerating sequence...
Ekhad, Shalosh B., Zeilberger, Doron
We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna...
Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico (1995)
Dodgson's determinant condensation rule is shown to immediately imply the evaluation of MacMahon's determinant expression that leads to the Box Theorem.
Fred Galvin's amazing proof of the Dinitiz conjecture is used to illustrate the method of undetermined generalization and specialization.
Self Avoiding Walks, the Language of Science, and Fibonacci Numbers (1995)
The Bordelaise philosophy, or rather a juvenile version of it, is used to enumerate self avoiding walks in a $[0,1] \times (- \infty, \infty)$.
Graphical major indices (1995)
Dominique Foata, Doron Zeilberger
Abstract: A generalization of the classical statistics "maj " and "inv " (the major index and number of inversions) on words is introduced, parameterized by...
Graphical major indices (1995)
Dominique Foata, Doron Zeilberger
Abstract: A generalization of the classical statistics "maj " and "inv " (the major index and number of inversions) on words is introduced, parameterized by...
Proof Of The Alternating Sign Matrix Conjecture (1995)
Doron Zeilberger, Gert Almkvist, Noga Alon, George Andrews, Dror Bar-natan, Francois Bergeron, ...
: The number of n n matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved...
Shalosh B. EKHAD, Doron Zeilberger
: We prove an explicit formula for the number of n \Theta n upper triangular matrices, over GF (q), whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna...
Proof of the Alternating Sign Matrix Conjecture (1994)
The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is...
Ekhad, Shalosh B., Zeilberger, Doron
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
Three Recitations on Holonomic Systems and Hypergeometric Series (1994)
A tutorial on what later became to be known as WZ theory, as well as a motivated account of the seminal Gosper algorithm.
The Graphical Major Index (1994)
Foata, Dominique, Zeilberger, Doron
A generalization of the classical statistics ``maj'' and ``inv'' (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The...
How Joe Gillis Discovered Combinatorial Special Function Theory (1994)
How Enumerative Combinatorics met Special Functions, thanks to Joe Gillis
journal of statistical planning and inference Self-avoiding walks, the language of science,
Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory (1993)
Foata, Dominique, Zeilberger, Doron
Capelli's and Turnbull's classical identities are given elegant combinatorial proofs.
A high-school algebra wallet-sized proof, of the Bieberbach conjecture After L. Weinstein] (1993)
Ekhad, Shalosh B., Zeilberger, Doron
Weinstein's[2] brilliant short proof of de Branges'[1] theorem can be made yet much shorter(modulo routine calculations), completely elementary (modulo L\"owner theory), self contained(no need for...
The Vandermonde-Chu Binomial Coefficients Identity is shown to imply Bombieri's deep norm inequalities, via identities of Beauzamy-D\'egot, and Reznick.
A WZ proof of Ramanujan's Formula for Pi (1993)
Ekhad, Shalosh B., Zeilberger, Doron
Ramanujan's series for Pi, that appeared in his famous letter to Hardy, is given a one-line WZ proof.
Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture (1993)
The future of mathematics is described, by using the WZ algorithmic proof theory as a parable.
George E. Andrews, Shalosh B. Ekhad, Doron Zeilberger
n\Gammak = 1; (a) n X k=0 2(\Gammaq n+1 ) k 1 + q k H k H n = n X k=\Gamman (\Gammaq) k 2 : (b) Proof: Let L 1 (n) and L 2 (n) be the left sides of (a) and (b) respectively, and let F 1 (n; k), and F...
Jane Friedman, Ira Gessel T, Doron Zeilberger, Communicated George Andrews
Two are homing a talit, one is saying it is all his, and (the other) one is saying it is all his.... let them each get half.--BAvA METZIA Consider all planar walks, with positive unit steps (1, 0)...
Rational function certification of multisum/integral/``$q$'' identities (1992)
Wilf, Herbert S., Zeilberger, Doron
The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and ``$q$'' integral/sums.
Andrews, George, Ekhad, Shalsoh B., Zeilberger, Doron
A short and elementary proof, and a finite-form generalization, are given of Jacobi's formula for the number of ways of writing an integer as a sum of four squares (that implies Lagrange's famous...
Rational function certification of multisum /integral/"q" identites (1992)
Herbert S. Wilf, Doron Zeilberger
Abstract. The method of rational function certification for proving terminating hypergeometric identities is extended from single sums or integrals to multi-integral/sums and “q ” integral/sums....
Conjectured Forrester, Doron Zeilberger
As always in q-theory, (X;Q)n will stand for the product (1-X)(1-QX)...(1-Qn-IX), and when the "base " Q is q, we will abbreviate (X; q)n to (X)n. For any Laurent polynomial f in x~,...,...
Herbert S. Wilf, Doron Zeilberger
this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term...
JOURNAL OF COMBINATORIAL THEORY, Series A 66, 17-27 (1994) A Constant Term Identity (1991)
Andrews's recent proof of the Mills-Robbins-Rumsey conjectured formula for the number of totally symmetric self-complementary plane partitions is used to derive a new multi-variate constant term...
The method of creative telescoping (1991)
An algorithm for de6nite hypergeometric summation is given. It is based, in a non-obvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification...
The method of differentiating under the integral sign (1990)
Gert Almkvist, Doron Zeilberger
”I could never resist a definite integral ” (G. H. Hardy [Co]) ”One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my...
Rational functions certify combinatorial identities (1990)
Herbert S. Wilf, Doron Zeilberger
This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present acerti cate that consists of a pair of functions of two integer...
Rational Functions Certify Combinatorial Identities (1989)
Herbert S. Wilf, Doron Zeilberger
This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present a certificate that consists of a pair of functions of two integer...
In his celrhra~cd paper, Polya has considered rht. random walk in thc threedimensional (cubic) lattice and showed that the probability of return to the orig~n is less than 1. Suhsequcni authors have...
In accordance with the principle from other branches of mathematics that it is better to exhibit an explicit isomorphism between two objects than merely to prove that they are isomorphic, we adopt...
Proof of Dougall’s Sum Identity (1988)
Shalosh B. Ekhad, Doron Zeilberger, C. Berndt
My guess is that, within fifty or hundred years (or it might be one hundred and fifty) computers will successfully compete with the human brain in doing mathematics, and that their mathematical style...
Linearization Coefficients For The Jacobi Polynomials (1987)
Dominique Foata, Doron Zeilberger
. --- The explicit non-negative representation of the linearization coefficients of the Jacobi polynomials obtained by RAHMAN seems to be difficult to be derived by combinatorial methods. However...
Linearization coefficients for the Jacobi polynomials (1987)
Dominique Foata, Doron Zeilberger
RÉSUMÉ. — Une formule explicite pour les coefficients de linéarisation des polynômes de Jacobi a été donnée par RAHMAN, d’où l’on tire, sans calcul, les propriétés de positivité....
Once on the forefront of mathematical research in America, the asymptotics of the solutions of linear recurrence equations is now almost forgotten, especially by the people who need it most, namely...
NOTE ENUMERATING TOTALLY CLEAN WORDS (1985)
Let A be a finite alphabet and let D be a finite set of words in A * labelled dirty. We give a recursive procedure for computing the generating function for the number of words not containing any...
Weighted Derangements And Laguerre Polynomials (1984)
Dominique Foata, Doron Zeilberger
this paper to provide one by taking up again the combinatorial model introduced by GILLIS and EVEN [7] and "ff-extending" it. Let P(n 1 ; : : : ; nm ) be the set of permutations on the n 1...
Weighted derangements and Laguerre polynomials (1984)
Dominique Foata, Doron Zeilberger
of the theory of special functions was a remarkable result of GILLIS and EVEN [7] that gave a certain combinatorial interpretation to the linearization coefficients of the simple Laguerre polynomials...
André’s reflection proof generalized to the many-candidate ballot problem (1983)
• • • There is a war between the odd and ~he even. (Leonard Cohere) Candidates 1...., n received mr,,.., m, ~ votes respectively, where rn ~>t. • • ~> m, # 0. The ballot problem asks...
Sister Celine's technique and its generalizations (1982)
Sister Celine Fasenmyer’s technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and generalized in various directions. Applications include algorithms for...
A Markov Chain Occurring in Enzyme Kinetics (1982)
Abstract. A certain Markov chain which was encountered by T. L. Hill in the study of the kinetics of a linear array of enzymes is studied. An explicit formula for the steady state probabilities is...
The algebra of linear partial difference operators and its applications (1980)
Doron Zeilberger, Dedicated Richard, J. Duffin
Abstract. The algebra of linear partial difference operators is investigated, and an elimination procedure demonstrated. Applications to combinatorics are given. In particular, a new proof and a...
Solutions of exponential growth to systems of partial differential equations (1979)
Integral representation formulas are established for functions of exponential growth satisfying a system of homogeneous partial differential equations with constant coefficients. This is a special...
Dym: Further properties of discrete analytic functions (1977)
Analogs of the classical theorems of Liouville, PhragmCn-LindeEf, and Paley-Wiener are proved in the class of discrete analytic functions. 1.
Superficially, this article, dedicated with friendship and admiration to Amitai Regev, has nothing to do with either Polynomial Identity Rings, Representation Theory, or Young tableaux, to all of...
All binomial identities are verifiable
Sister Celine Fasenmyer's technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and used to show that every identity involving sums of products of binomial...
Pompeiu's problem on discrete space
Let [unk] be a finite family of finite subsets of the n-dimensional lattice Zn, and let τ denote the group of all translations of Zn. We shall here consider the Pompeiu problem for the family...
All binomial identities are verifiable
Sister Celine Fasenmyer's technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and used to show that every identity involving sums of products of binomial...
Pompeiu's problem on discrete space
Let [unk] be a finite family of finite subsets of the n-dimensional lattice Zn, and let τ denote the group of all translations of Zn. We shall here consider the Pompeiu problem for the family...
The quantum MacMahon Master Theorem
Garoufalidis, Stavros, Lê, Thang T. Q., Zeilberger, Doron
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.