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Shalosh B. Ekhad

Publications and source records attributed to Shalosh B. Ekhad.

At least 73 records · Page 4Linked to original sources
arXiv2015

Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution

Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of hook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (and Paul Johnson, and a bit later, Victor Wang, proved) a beautiful expression for the average size, namely (s-1)(t-1)(s+t+1)/24 . In the present article, we go far beyond the average, and state absolutely certain expressions (but "officially" still conjectures) for the variance (showing in particular that it is rather large, and there is no "concentration about the mean"), and the third through the sixth moments. For the special case of (s,s+1)-core partitions, we go all the way to the 9th moment. We pose two challenges, and will be glad to donate 100 dollars each, to the OEIS foundation in honor of the first provers, regarding a "soft" and "global", yet rigorous, justification of our empirical approach, and for proving an intriguing conjecture about the limiting distribution. This version reports (thanks to Marko Thiel and Nathan Williams) that the second challenge mentioned above has been done by Paul Johnson (but not in two pages). A donation to the OEIS was made. The first challenge is still wide open.

math.CO↗
arXiv2015

The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r

We use the Robinson-Schensted correspondence, followed by symbol-crunching, in order to derive explicit expressions for the quantities mentioned in the title. We follow it by number crunching, in order to compute the first terms of these sequences. As an encore, we cleverly implement Ira Gessel's celebrated determinant formula for the generating functions of these sequences, to crank out many terms. This modest tribute is dedicated to one of the greatest enumerators alive today (and definitely the most modest one!), Ira Martin Gessel, who is turning 64 years-old today

math.CO↗
arXiv2015

Odd-Rule Cellular Automata on the Square Grid

An "odd-rule" cellular automaton (CA) is defined by specifying a neighborhood for each cell, with the rule that a cell turns ON if it is in the neighborhood of an odd number of ON cells at the previous generation, and otherwise turns OFF. We classify all the odd-rule CAs defined by neighborhoods which are subsets of a 3 X 3 grid of square cells. There are 86 different CAs modulo trivial symmetries. When we consider only the different sequences giving the number of ON cells after n generations, the number drops to 48, two of which are the Moore and von Neumann CAs. This classification is carried out by using the "meta-algorithm" described in an earlier paper to derive the generating functions for the 86 sequences, and then removing duplicates. The fastest-growing of these CAs is neither the Fredkin nor von Neumann neighborhood, but instead is one defined by "Odd-rule" 365, which turns ON almost 75% of all possible cells.

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arXiv2015

The Method(!) of "Guess and Check"

The problems of enumerating lattice walks, with an arbitrary finite set of allowed steps, both in one and two dimensions, where one must always stay in the non-negative half-line and quarter-plane respectively, are used, as case studies, to illustrate the `naive' methodology of guess-and-check, where rigorous proofs are possible, but not worth the trouble. We argue that this is a metaphor for future math.

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arXiv2014

The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ... , n are D-finite for all d and all r

In this article, dedicated with admiration and gratitude to guru Neil Sloane on his 75-th birthday, we observe that the generating functions for multi-set permutations that do not contain an increasing subsequence of length d, and where every letter appears the same number of times, say r, are always D-finite, (for every d and every r), and we actually crank out the first few terms of quite a few of them, many of whom are not yet in the OEIS. We also state a conjectured asymptotic formula for these sequences, that reduces to Amitai Regev's famous formula when r=1, and pledge a 100 dollar donation to the OEIS in honor of the first one to prove our conjecture. We pledge another 100 dollars for extending Ira Gessel's spectacular Bessel determinant, from the r=1 case to general r.

math.CO↗
arXiv2014

How to Gamble If You're In a Hurry

The beautiful theory of statistical gambling, started by Dubins and Savage (for subfair games) and continued by Kelly and Breiman (for superfair games) has mostly been studied under the unrealistic assumption that we live in a continuous world, that money is indefinitely divisible, and that our life is indefinitely long. Here we study these fascinating problems from a purely discrete, finitistic, and computational, viewpoint, using Both Symbol-Crunching and Number-Crunching (and simulation just for checking purposes).

math.PR↗
arXiv2014

A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage)

Once upon a time there was an esoteric and specialized notion, called "size of the Durfee square", of interest to at most 100 specialists in the whole world. Then it was kissed by a prince called Jorge Hirsch, and became the famous (and to quite a few people, infamous) h-index, of interest to every scientist, and scholar, since it tells you how productive a scientist (or scholar) you are! When Rodney Canfield, Sylvie Corteel, and Carla Savage wrote their beautiful 1998 article proving, rigorously, by a very deep and intricate analysis, the asymptotic normality of the random variable "size of Durfee square" defined on integer-partitions of n (as n goes to infinity), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to everyone who has ever published a paper. However Canfield et. al. had to work really hard to prove their deep result. Here we take an "empirical" shortcut, that proves the same thing much faster (modulo routine number- and symbol- crunching). More importantly, the empirical methodology should be useful in many other cases where rigorous proofs are either too hard, or not worth the trouble!

math.HO↗
arXiv2014

There are $(r+1)(r+2)(2r+3)(r^2+3r+5)$ Ways For the Four Teams of a World Cup Group to Each Have $r$ Goals For and $r$ Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1]

This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when one the authors(DZ) attended the Stanely@70 conference, that took place at the same time as the preliminary stage of the 2014 World Cup. It states a surprising application of an analog of Richard Stanley's famous theorem about the enumeration of magic squares to the enumeration of possible outcomes in a World Cup Group.

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arXiv2014

Enumerative Geometrical Genealogy (Or: The Sex Life of Points and Lines)

Joshua Cooper and Mark Walters initiated the subject that we call here Enumerative Geometric Genealogy, (and that lead to the hard-to-count OEIS sequence A140468) but they did not realize the broader impact of their research to the population explosion of a certain far-away planet, whose inhabitants are much smarter, and more civilized, than we are.

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arXiv2014

Automatic Proofs of Asymptotic ABNORMALITY (and much more!) of Natural Statistics Defined on Catalan-Counted Combinatorial Families

In this case-study in computer-human collaboration, we develop, implement, and execute symbolic-computational algorithms for the automatic discovery and proof of explicit expressions for the expectation, variance, and higher moments of a large class of natural combinatorial statistics defined on Catalan-counted objects, enabling, inter-alia, to prove that they are not asymptotically normal. In particular, we reproduce in 0.12 seconds results of Miklos Bona, and derive far deeper results, way beyond the scope of humans, concerning higher moments of the random variable "number of occurrences of a pattern" in the set of 132-avoiding permutations for all patterns of length 2 and 3, and, more impressively, explicit expressions for the averages for all patterns of lengths up to 10. The ample output inspired us to make an intriguing conjecture concerning the number of so-called Bona classes, and we pledge to donate 100 dollars to the OEIS Foundation in honor of the prover (or disprover).

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arXiv2013

How to Extend Karolyi and Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem in order to Evaluate ANY Coefficient of the q-Dyson Product

We show how to extend the Karolyi-Nagy beautiful proof of the Zeilberger-Bressoud q-Dyson theorem, (first proved by Zeilberger and Bressoud in 1985, and originally conjectured by George Andrews in 1975), that states that the constant term of a certain Laurent polynomial equals the q-multinomial coefficient, how to evaluate any other specific coefficient. The algorithm implies that any such coefficient is always a certain rational function (that the algorithm finds) times the q-multinomial coefficient.

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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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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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