arXiv subjects
Publications and source records attributed to Doron Zeilberger.
We use the Almkvist-Zeilberger algorithm, combined with a weighted version of the Even-Gillis Laguerre integral due to Foata and Zeilberger, in order to efficiently compute weight enumerators of multiset derangements according to the number of cycles. The present paper is inspired by important previous work by Mourad Ismail and his collaborators, done in the late 1970s, but still useful after all these years.
We give a very short proof of the fact that the number of $(a,b)$-parking functions of length $n$ equals $a(a+bn)^{n-1}$. This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general result. This new version contains a reference to previous work kindly communicated by Richard Stanley
Using Symbolic Dynamic Programming we describe algorithms, fully implemented in Maple, for automatically generating generating functions introduced by Richard Stanley in his study of generalized Stern arrays, generalized even further, to arrays defined in terms of general sequences satisfying linear recurrences with constant coefficients, rather than just the Fibonacci and k-bonacci sequences
One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an $O(1/n^{85})$ asymptotic formula.
Twenty years ago, Werner Krandick defined two statistics on binary trees. The first one determines the number of jumps, when traversing the tree in depth-first-search, from a vertex to one closer to the root, and the second keeps tracks of the sum of the jump-distances. He used clever but ad hoc human-generated arguments to find explicit expressions for their expectations. In this methodological note, we illustrate the power of experimental mathematics and symbolic computation to do much more. We derive closed-form expressions for the actual weight-enumerators according to these statistics (from which not only the expectations, but also the variances, and as many higher moments as desired, can be obtained). We also actually give the first eight moments, and conjecture that the first statistic (number of jumps) is asymptotically normal, and prove that the second one (sum of jump distances) is definitely not. In this revised version we are happy to announce that Stephen Melczer and Tia Ruza fully proved the asymptotic normality, and we provide a link to their writeup.
In 1750, Gabriel Cramer famously stated, without proof, his eponymous rule for solving a system of linear equations with as many equations as unknowns. We give a purely combinatorial, and purely self-contained, proof of this old chestnut, still useful after all these years.
John von Neumann studied a simplified version of poker where the "deck" consists of infinitely many cards, in fact, all real numbers between $0$ and $1$. We harness the power of computation, both numeric and symbolic, to investigate analogs with finitely many cards. We also study finite analogs of a simplified poker introduced by D.J. Newman, and conclude with a thorough investigation, fully implemented in Maple, of the three-player game, doing both the finite and the infinite versions. This paper is accompanied by two Maple packages and numerous output files.
We introduce a very simple solitaire game, named Stanley Solitaire, in honor of Richard Stanley, and prove an explicit closed-form formula for the number of ways of playing it. Alas, the only proof that we know is via a deep theorem of Richard Stanley from 1984. We challenge the readers to find a more elementary proof.
Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient polynomial $p(x, y, z)$ whose only integer roots are consecutive triples of Tribonacci numbers.
On March 16, 2024, Daniel Litt, in an X-post, proposed the following brainteaser: "Flip a fair coin 100 times. It gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, so e.g. for the sequence THHHT Alice gets 2 points and Bob gets 1 point. Who is most likely to win?" We show the power of symbolic computation, in particular the (continuous) Almkvist-Zeilberger algorithm, to answer this, and far more general, questions of this kind.
Starting from AJ Bu's recent article that computed explicit expressions for the GENERATING functions of sums of powers of areas under Dyck and Motzkin paths, we deduce from them explicit expressions for the actual sequences. This enables taking the limits of the scaled moments and confirming, in an entirely elementary way, that they tend to those of the area under Brownian Excursion (up to any specified moment).
We present both bijective and automated approaches to Abel-type sums, dear to Dominique Foata.
We use Experimental Mathematics and Symbolic Computation (with Maple), to search for lots and lots of Perrin- and Lucas- style primality tests, and try to sort the wheat from the chaff. More impressively, we find quite a few such primality tests for which we can explicitly construct infinite families of pseudo-primes, rather, like in the cases of Perrin pseudo-primes and the famous Carmichael primes, only proving the mere existence of infinitely many of them.
In his seminal paper ``A census of planar triangulations", published in 1962, the iconic graph theorist (and code-breaker), W.T. Tutte, spent a few pages to prove that a certain bi-variate generating function that enumerates triangulations, satisfies a certain functional equation. He then used his genius to actually solve it, giving closed-form solutions to the enumerating sequences. While the first part, of deriving the functional equation, still needs human ingenuity, the second part, of solving it, can nowadays be fully automated. Our Maple program, accompanying this paper, Tutte.txt, can not only solve Tutte's original equation in a few seconds, it can also solve many, far more complicated ones, way beyond the scope of even such a giant as W.T. Tutte. We use our favorite method of ``guess and check" and show how it can always be made fully rigorous (if desired).
In their study of water waves, Massimiliano Berti, Livia Corsi, Alberto Maspero, and Paulo Ventura, came up with two intriguing conjectured identities involving certain weighted sums over the Boolean lattice. They were able to prove the first one, while the second is still open. In this methodological note, we will describe how to generate many terms of these types of weighted sums, and if in luck, evaluate them in closed-form. We were able to use this approach to give a new proof of their first conjecture, and while we failed to prove the second conjecture, we give overwhelming evidence for its veracity. In this second version, we are happy to announce that Mark van Hoeij was able to complete the proof of the second conjecture, by explicitly solving the second-order recurrence mentioned at the end.
In their recent claimed computer-free proof of the Four Color Theorem, David Jackson and Bruce Richmond attempted to use sophisticated "asymptotic analysis" to explicitly compute a certain number whose positivity (according to them) implies this famous theorem. While the jury is still out whether their valiant attempt holds water, we prove, in this modest note, that this constant equals exactly 10/27. We also point out that their evaluation of this constant must be erroneous, for two good reasons. Finally, as an encore, we state many similar, but more complicated, results.
We study maximal seating arrangements, either on a line, or in a rectangular auditorium with a fixed number of columns but an arbitrary number of rows, that obey any prescribed set of `social distancing' restrictions. In addition to enumeration, we study the statistical distribution of the density, and give efficient algorithms for generating these at random
We revisit, implement, and experiment with a beautiful algorithm, due to Calabi and Wilf for the random generation of subspaces over a finie field.