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Shalosh B. Ekhad

Publications and source records attributed to Shalosh B. Ekhad.

At least 37 records · Page 2Linked to original sources
arXiv2022

Linear-Time and Constant-Space Algorithms to compute Multi-Sequences that arise in Enumerative Combinatorics (and Elsewhere)

How many ways, exactly, can a Chess King, always moving forward (i.e. with steps [1,0],[0,1],[1,1]) walk to [100000,200000]? Thanks to the amazing Apagodu-Zeilberger extension of the Almkvist-Zeilberger algorithm, adapted in this article for combinatorial applications, this 104492-digit number, can be computed in less than 33 seconds. But not just this particular number. Many other numbers that come up in enumerative combinatorics, can be computed just as efficiently

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arXiv2022

Automated Counting and Statistical Analysis of Labeled Trees with Degree Restrictions

Arthur Cayley famously proved that there are n to the power n-2 labeled trees on n vertices. Here we go much further and show how to enumerate, fully automatically, labeled trees such that every vertex has a number of neighbors that belongs to a specified finite set, and also count trees where the number of neighbors is not allowed to be in a given finite set. We also give detailed statistical analysis, and show that in the sample space of labeled trees with n vertices, the random variable "number of vertices with d neighbors" is asymptotically normal, and for any different degrees, are jointly asymptotically normal, but of course, not independently so (except for the pair (1,3), i.e. the number of leaves and the number of degree-3 vertices, where there are asymptotically independent). We also give new proofs to Amram Meir and John Noon's expressions for the limiting expectation and variance for these, and derive an explicit expression for the covariance.

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arXiv2022

A Linear Time, and Constant Space, Algorithm to Compute the Mixed Moments of the Multivariate Normal Distributions

Using recurrences gotten from the Apagodu-Zeilberger Multivariate Almkvist-Zeilberger algorithm we present a linear-time, and constant-space, algorithm to compute the general mixed moments of the k-variate general normal distribution, with any covariance matrix, for any specific k. Besides their obvious importance in statistics, these numbers are also very significant in enumerative combinatorics, since they count in how many ways, in a species with k different genders, a bunch of individuals can all get married, keeping track of the different kinds of heterosexual marriages. We completely implement our algorithm (with an accompanying Maple package, MVNM.txt) for the bivariate and trivariate cases (and hence taking care of our own 2-sex society and a putative 3-sex society), but alas, the actual recurrences for larger k took too long for us to compute. We leave them as computational challenges.

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arXiv2021

Automating John P. D'Angelo's method to study Complete Polynomial Sequences

In this article, dedicated to the memory of Ron Graham, we fully automate John P. D'Angelo's method for studying complete polynomial sequences, for which Ron Graham, back in 1964, gave a beautiful necessary and sufficient conditions. D'Angelo found surprising connection of these elementary (but deep) number-theoretic questions to determining certain critical dimensions in complex geometry.

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arXiv2021

An Experimental (yet fully rigorous!) Study of a certain "Measure Of Disarray" that 12-year Noga Alon Proved was always Even

We study in depth a certain permutation statistic that was the subject of a brilliant insight by 12-year-old Noga Alon. Our approach is purely empirical and experimental, yet it is fully rigorous, thereby debunking, yet another time, the myth that mathematics is always a deductive science. This revised version contains three postscripts describing improvements pointed out by Stoyan Dimitrov, Kyle Petersen, and Martin Rubey.

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arXiv2021

Automatic Generation of Convolution Identities for C-finite sequences

In a recent insightful article, Helmut Prodinger uses sophisticated complex analysis, with residues, to derive convolution identities for Fibonacci, Tribonacci, and k-bonacci numbers. Here we use a naive, "experimental mathematics" (yet fully rogorous!) approach, using the C-finite ansatz, that can derive such identities in a few seconds, but not just for the above-mentioned sequences, but for every C-finite sequence (i.e. a sequence satisfying a linear recurrence with constant coefficients), and even for a pair of these.

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arXiv2021

On Invariance Properties of Entries of Matrix Powers

A few years ago, Peter Larcombe discovered an amazing property regarding two by two matrices. For any such 2 by 2 matrix A, the ratios of the two anti-diagonal entries is the same for all powers of A. We discuss extensions to higher dimensions, and give a short bijective proof of Larcombe and Eric Fennessey's elegant extension to tri-diagonal matrices of arbitrary dimension. This article is accompanied by a Maple package.

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arXiv2021

Some Deep and Original Questions about the "critical exponents" of Generalized Ballot Sequences

We numerically estimate the critical exponents of certain enumeration sequences that naturally generalize the famous Catalan and super-Catalan sequences, and raise deep and original questions about their exact values, and whether they are rational numbers. In this version we announce that our questions were brilliantly answered by Michael Wallner, and that the pledged donation to the OEIS, in his honor, was made.

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arXiv2021

A One-Line Proof of Leversha's "Quartet of Isogonal Conjugates" Theorem

We give a short and insightful proof of Gerry Leversha's elegant theorem regarding the isogonal conjugates of each of the vertices of a non-cyclic quadrilateral with respect to the triangle formed by the other three. It uses the Maple package RENE.txt, available from . http://www.math.rutgers.edu/~zeilberg/tokhniot/RENE.txt

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arXiv2021

A Bijective Proof of Richard Stanley's Observation that the sum of the cubes of the n-th row of Stern's Diatomic array equals 3 times 7 to the power n-1

In a delightful article, Richard Stanley derived, algebraically, the surprisingly simple formula, 3 times 7 to the power n-1, for the sum of the cubes of the n-th row of Stern's diatomic array. In this note, we find an elegant bijective proof of this surprising fact, that explains it and gives insight. The novelty is that this gorgeous bijection was discovered by a computer (SBE), with minimal guidance by a human (DZ). This debunks the conventional wisdom, held by some human supremacists, that computers can only compute, but they can't give insight

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arXiv2021

There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]

In this memorial tribute to Joe Gillis, who taught us that Special Functions count, we show how the seminal Even-Gillis integral formula for the number of derangements of a multiset, in terms of Laguerre polynomials, can be used to efficiently compute not only the number of the title, but much harder ones, when it is interfaced with Wilf-Zeilberger algorithmic proof theory.

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arXiv2020

Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan

In Ramanujan's Lost Notebook there is an amazing identity that furnishes infinitely many "almost counterexamples" to the cubic Fermat's Last Theorem, with no indication whatsoever how he discovered it. In 1995, Michael Hirschhorn explained, in a brilliant way, how Ramanujan may have done it, based on a certain polynomial identity for a sum of four cubes. Much earlier, Eri Jabotinsky, in an article published in 1946 (in a mathematics journal for teenagers) explained how Ramanujan may have discovered these polynomial identities needed for Hirschhorn's approach. Here we combine these two brilliant ideas (that may or may not have been how Ramanujan did it), automate it, and generalize, by developing an algorithm to solve a large class of cubic diophantine equations. Our interest in this problem was rekindled after reading Amy Alznauer's (b. Andrews) delightful children book "The Boy Who Dreamed of Infinity" (Candlewick Press), 2020, where Ramanujan's identity appears in one of the illustrations.

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arXiv2020

Almost avoiding permutations

We investigate the notion of almost avoiding a permutation: $π$ almost avoids $β$ if one can remove a single entry from $π$ to obtain a $β$-avoiding permutation.

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arXiv2020

Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming

Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families, using both a "dumb" approach (driven by numeric dynamic programming) that often works in practice, and a "clever" approach, needed for larger problems, driven by "symbolic" dynamic programming. Both approaches are fully automated and implemented in Maple.

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arXiv2020

The Absent-Minded Passengers Problem via Computer Algebra

In a delightful article that recently appeared in the American Mathematics Monthly, Norbert Henze and Guenter Last discuss the "Absent-Minded Passengers" Problem, but left open finding an explicit expression for the probability generating function, of the random variable "Number of passengers occupying a wrong seat", when the number of absent-minded passengers is larger than one. This is accomplished in this note, using experimental mathematics and symbolic computation. We also derive explicit expressions for the first 8 moments of the original case of one absent-minded passenger, and indicate how to extend it to the general case

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