1,151 research outputs found
Automated Counting of Towers (\`A La Bordelaise) [Or: Footnote to p. 81 of the Flajolet-Sedgewick Chef-d'{\oe}vre]
The brilliant idea of Jean Betrema and Jean-Guy Penaud that proved the
celebrated "three to the power n" theorem of Dominique Gouyou-Beauchamps and
Xavier Viennot, counting towers of domino pieces is extended and used to
enumerate much more general towers, where the pieces can be many i-mers.Comment: Accompanied by the Maple package TOWERS available from
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/migdal.html . That
page also has many deep computer-generated theorem obtained from the packag
A Treatise on Sucker's Bets
In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with
a set of four dice, let's call them A,B,C,D, whose faces are marked with
[0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5] respectively, where
die A beats die B, die B beats die C, die C beats die D, but, surprise
surprise, die D beats die A! This was an amazing demonstration that "being more
likely to win" is not a transitive relation. But that was only one example, and
of course, instead of dice, we can use decks of cards, where they are called
(by Martin Gardner, who popularized this way back in 1970) , "sucker's bets".
Can you find all such examples, with a specified number of decks, and sizes?
If you have a computer algebra system (in our case Maple), you sure can!
Not only that, we can figure out how likely such sucker bets are, and derive,
fully automatically, statistical information!Comment: 13 pages; Accompanied by two Maple packages and numerous input and
output files available from
http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/suckerbets.htm
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.Comment: 14 pages, accompanied by four Maple packages obtainable from
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gac.htm
The C-finite Ansatz Meets the Holonomic Ansatz
We show how the continuous Almkvist-Zeilberger algorithm can be used to
efficiently discover and prove differential equations satisfied by generating
functions of sequences defined as integrals of powers of C-finite polynomial
sequences (like the Chebyshev polynomials) from which one can automatically
derive linear recurrences with polynomial coefficients for the sequences
themselves. We fully implement this with a Maple package, CfiniteIntergal.txt .Comment: 4 pages. Accompanied by a Maple package, and sample input and output
files available from
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cfiniteI.htm
Some Remarks on a recent article by J. -P. Allouche
In 1980 Otto G. Ruehr made some puzzling comments that a certain identity A,
that he proved, is equivalent to another identity B, but he did not explain why
they are equivalent. Recently J.-P. Allouche tried to explain why they are
"equivalent", using about eight pages. We comment that it is extremely unlikely
to be Ruehr's reasoning (he probably got mixed up with a different problem),
but be that as it may, it is not worthwhile to try and deduce B from A, since
both A and B are routinely provable using Wilf-Zeilberger algorithmic proof
theory (implemented in Maple) and the Almkvist-Zeilberger algorithm.Comment: 4 page
How Many Rounds Should You Expect in Urn Solitaire?
A certain sampling process, concerning an urn with balls of two colors,
proposed in 1965 by B.E. Oakley and R.L. Perry, and discussed by Peter Winkler
and Martin Gardner, that has an extremely simple answer for the probability,
namely the constant function 1/2, has a far more complicated expected duration,
that we discover and sketch the proof of. So unlike, for example, the classical
gambler's ruin problem, for which both `probability of winning' and `expected
duration' have very simple expressions, in this case the expected number of
rounds is extremely complicated, and beyond the scope of humans.Comment: 5 pages, accompanied by Maple packages and output files available
from http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/urn.htm
A high-school algebra wallet-sized proof, of the Bieberbach conjecture After L. Weinstein]
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 esoteric Legendre polynomials'
addition theorem), and motivated(ditto), as follows
Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!)
In this short article, two recent beautiful proofs of George Andrews, Aviezri
Fraenkel, and James Sellers, about the mod m characterization of the number of
m-ary partitions are simplified and streamlined, and then generalized to handle
many more cases, and prove much deeper theorems, with the help of computers, of
course.Comment: 7 pages, accompanied by a Maple package, available from
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/afs.htm
On the number of Singular Vector Tuples of Hyper-Cubical Tensors
Shmuel Friedland and Giorgio Ottaviani's beautiful constant term expression
for the number of singular vector tuples of generic tensors is used to derive a
rational generating function for these numbers, that in turn, is used to obtain
an asymptotic formula for the number of such tuples for n by n by n
three-dimensional tensors, and to conjecture an asymptotic formula for the
general d-dimensional case. A donation of 100 dollars, in honor of the first
prover, will be made to the On-line Encyclopedia of Integer Sequences.Comment: 4 pages. Accompanied by a Maple package, SVT.txt available from
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/svt.htm
The "Monkey Typing Shakespeare" Problem for Compositions
Suppose that your mother gave you n candies. You have to eat at least one
candy each day. One possibility is to eat all n of them the first day. The
other extreme is to make them last n days, and only eat one candy a day.
Altogether, you have, famously, 2 to the power n-1 choices. If each such choice
is equally likely, what is the probability that you never have three
consecutive days, where in the first day you ate at least 2 candies, in the
second day you ate at least 5 candies, and in the third day you ate at least 3
candies? This article describes algorithms, fully implemented in two Maple
packages, to answer such important questions, and more general ones, of this
kind.Comment: 15 pages. Accompanied my two Maple packages available from
http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/kof.htm
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