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Doron Zeilberger

Publications and source records attributed to Doron Zeilberger.

At least 163 records · Page 9Linked to original sources
arXiv2013

Zeroless Arithmetic: Representing Integers ONLY using ONE

We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer.

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arXiv2013

A Short Proof of a Ptolemy-Like Relation for an Even number of Points on a Circle Discovered by Jane McDougall

We give a short proof of a Ptolemy-style result first discovered and proved by Jane McDougall. It may be viewed as a generalization to any even number of points of the cubic relation connecting the six joint distances of four points on a circle (whose quadratic relation is the famed Plucker relation, alias Ptolemy's theorem). This article is in fond memory of Andrei Zelevinsky (1953-2013) who loved Ptolemy's theorem.

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arXiv2013

On Euler's "Misleading Induction", Andrews' "Fix", and How to Fully Automate them

One of the greatest experimental mathematicians of all time was also one of the greatest mathematicians of all time, the great Leonhard Euler. Usually he had an uncanny intuition on how many "special cases" one needs before one can formulate a plausible conjecture, but one time he was "almost fooled", only to find out that his conjecture was premature. In 1990, George Andrews found a way to "correct" Euler. Here we show how to generate, AUTOMATICALLY, rigorously-proved Euler-Andrews Style formulas, that enables one to generate Euler-style "cautionary tales" about the "danger" of using naive empirical induction. Ironically, the way we prove the Andrews-style corrections is empirical! But in order to turn the empirical proof into a full-fledged rigorous proof, we must make sure that we check sufficiently many (but still not that many!) special cases.

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arXiv2013

How To Generate As Many Somos-Like Miracles as You Wish

Jacobi said "man muss immer umkehren". And indeed it takes a genius like Michael Somos to take a specific non-linear recurrence, like a(n)=(a(n-1)a(n-3)+a(n-2)^2)/a(n-4), subject to a(1)=1, a(2)=1, a(3)=1, a(4)=1, and observe that surprise, surprise, they always generate integers. Then it takes other geniuses to actually prove this fact (and the more general so-called Laurent phenomenon). But let's follow Jacobi's advise and go backwards. Rather than try to shoot a target fifty meters away, and most probably miss it, let's shoot first, and then draw the bull'e eye. Then we are guaranteed to be champion target-shooters. So let's take a sequence of integers that manifestly and obviously only consists of integers, and ask our beloved computers to find non-linear recurrences satisfied by the sequence itself, or by well-defined subsequences.

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arXiv2013

The Reciprocal of $\sum_{n\geq 0}a^nb^n$ for non-commuting $a$ and $b$, Catalan numbers and non-commutative quadratic equations

The aim of this paper is to describe the inversion of the sum $\sum_{n\geq 0}a^nb^n$ where $a$ and $b$ are non-commuting variables as a formal series in $a$ and $b$. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.

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arXiv2012

Proof of George Andrews's and David Robbins's q-TSPP Conjecture

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.

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arXiv2012

Pick Up Sticks

We apply a classic result in Theoretical Computer Science to solve a sticky problem, and give a neat and slick quick proof of half of it

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arXiv2012

Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes

One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a "formula", or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate Functional Equations, alas, with an unbounded number of "catalytic variables", but then described a clever way, using multivariable calculus, how to get enumeration schemes. Alas, their method becomes very complicated for r larger than 1. In the present article we describe a far simpler way to squeeze the necessary information, in polynomial time, for increasing patterns of any length, and for any number of occurrences, r.

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arXiv2012

The Amazing $3^n$ Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang]

The most amazing (at least to me) result in Enumerative Combinatorics is Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the number of so-called directed animals with compact source (that are equivalent, via Viennot's beautiful concept of heaps, to towers of dominoes, that I take the liberty of renaming xaviers) with n+1 points equals 3^n. This amazing result received an even more amazing proof by Jean Bétrema and Jean-Guy Penaud. Both theorem and proof deserve to be better known! Hence this article, that is also accompanied by a comprehensive Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/BORDELAISE that implements everything (and much more)

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arXiv2012

Automatic Counting of Tilings of Skinny Plane Regions

The deductive method ruled mathematics for the last 2500 years, now it is the turn of the inductive method. Here we make a modest start by using the inductive method to discover and prove (rigorously) explicit generating functions for the number of dimer (and monomer-dimer) tilings of large families of "skinny" plane regions.

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arXiv2012

Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux

In how many ways can you place n chocolate pieces all of different sizes in an n by n chocolate box, in such a way that when you go from left to right and from top to bottom, there are no gaps AND the sizes increase along each row and each column? The answer is the well-known OEIS Sequence Number 85. To our amazement, the analogous sequence for a three-dimensional chocolate box was not there. Here we fill this gap, and more importantly, offer some computational and theoretical challenges about enumerating families of Solid Standard Young Tableaux.

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arXiv2012

Using GENERATINGFUNCTIONOLOGY to Enumerate Distinct-Multiplicity Partitions

This article, written in fond memory of Herbert Saul Wilf (June 13, 1931- Jan. 7, 2012), explores integer partitions where each part shows up a different number of times than the other parts (if it shows up at least once), thereby making a modest contribution towards the solution of one of the eight intrigiung problems posted by Herb Wilf on his website on Dec. 13, 2010

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arXiv2011

Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type

Richard Stanely proposed, in a recent Amer. Math. Monthly Problem, to prove a nice explicit formula for the generating function for the number of n-letter words in {H,T} that have as many occurrences of HT as HH. In this article, we show how to prove this problem automatically, and ANY problem of that type, regardless of the size of the alphabet and the length of the two chosen strings

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arXiv2011

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.

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arXiv2011

Rademacher's infinite partial fraction conjecture is (almost certainly) false

In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most $N$ parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher's conjectured limits for certain (predictable) indices in the sequences.

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arXiv2011

Alexander Burstein's Lovely Combinatorial Proof of John Noonan's Beautiful Formula that the number of n-permutations that contain the Pattern 321 Exactly Once Equals (3/n)(2n)!/((n-3)!(n+3)!)

In 1996, my brilliant student John Noonan, discovered, and proved that there are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one is such that the tallest stands (not necessarily immediately) in front of the second-tallest, who in turn, stands (not necessarily immediately) in front of the shortest. In that article, I promised a prize of 25 dollars for a nice combinatorial proof. Alex Burstein gave such a proof. On Oct. 14, 2011, I talked about Alex's lovely proof at the Howard U. math colloquium, and publicly presented the 25-dollar prize. The present note is the outcome. Congratulations Alex!

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