Disclaimer: some of the earlier progrmas may not run properly on the current version of Maple, since Maple is, unfortunately

Maple packages accompanying A=B

ZeroKnowledgeProof.txt,
A Maple implementation of Manuel Blum's
seminal article
"How to Prove a Theorem so that no one can claim it"
[Proceedings of the International Congress of Mathematicians, Berkely, California, USA, 1986,pages 1444-1451]

For a very lucid and engaging explanation of Zero Knowledge Proofs see
here.

The four Maple packages SnakesAndLadders.txt, PosPileGames.txt, GenPileGames.txt, VGPileGames.txt accompany the beautiful article, joint with Thotsaporn Aek Thanatipanonda.

NumberCrunch.txt, to create and solve "Number Crunch" probelms that appear in the Australian daily newspaper "The Australian". It generated this web-book.

The Maple package SlaterValez.txt, studies OEIS sequence A081145. It accompanies this article.

The Maple package BINnr.txt, studies throwing balls into bins It accompanies this article.

The Maple package QuickSortAnalysis.txt, automatically studies the distribution of running time of Quicksort. It accompanies this article.

The Maple package MagicRectangles.txt, automatically finds explicit expressions for the number of semi-magic rectangles of a given size. It accompanies this article.

The Maple package SINKHORN.txt, automatically computes explicit expressions for the Sinkhorn limits of matrices It accompanies this article.

The Maple package EvenChange.txt, automatically generate theorems enumerating number of ways of having a total of n coins in your two pockets with the same amount in both. It was inspired by a beautiful 1980 article of Gert Almkvist (1934-2018). It accompanies this article.

The Maple package Compositions.txt, and the more general one (that includes the former) CompositionsPlus.txt, automatically generate theorems enumerating compsitions that avoid (or contain a specified number of) a given finite set of compositions by containment. They accompany this article.

The three Maple packages GFMatrix.txt, SpanningTrees.txt, and JointConductance.txt, written with Yukun Yao, automatically generates generating functions for interesting and important enumeration sequences. They accompany this article.

Bitcoin.txt, a Maple porgram for the probabilistic analysis of bitcoin attacks, aka as two-phase soccer matches. It accompanies this article.

CHOMP.txt, a short porgram (using the obvious algorithms) to compute the winning moves (if they exist) for Chomp positions [inspired by email message from Purui Zhang and Lu Yan]. It accompanies this article.

Theta3Romik.txt,
a Maple package to implement and investigate Dan Romik's fascinating article
The Taylor coefficients of the Jacobi theta constant &theta_{3}.

To use it, see the
user's manual.

QuasiDeterminants.txt, a Maple package to handle quasi-determinants, dear to Gelfand, Volodia Retakh and many other people. dedicated to Vladimir Retakh on his 70th birthday.

The Maple package
ParkingStatistics.txt,
written with Yukun Yao, investigates the area statistics of parking functions.
It accompanies this article.

[A related, small package, just about parking functions, demonstating Henry Pollak's clever proof of
the fact that there are (n+1)^{(n-1)} parking functions of length n, is the Maple package
ParkingPollack.txt]

FOOL.txt, a Maple package to automatically produce books like this one, dedicated to George Andrews on his forthcoming 80th birthday.

MNTD.txt, a Maple package to implement the beautiful article by Joe Buhler, Ron Graham, and Al Hales, "Maximally Nontransitive dice" that appeared in the American Mathematical Monthly 125(5)( May 2018),387-399. For example the input file yields the output file.

MKP.txt, a Maple package to solve meta-knowldege puzzles inspired by the puzzle "Sums and Products" in Peter Winkler's book "Mathematical Mind-Benders" (p. 114). [If you use this input file, you would get this puzzle book (with 40 challenging puzzles, with answers at the bottom), and if you use this input file, you would get detailed solution to these 40 puzzles.]

UrnSolitaire.txt,
a Maple package to study the game of Urn Solitaire described in
Peter Winkler's book "Mathematical Mind-Benders" (p. 76).

It accompanies by this article.
[It requires the Maple package GuessHolo2.txt in the same directory.]

ROKHLENKO.txt, a short Maple package that is a computational appendix to an article by A. Rokhlenko and J.L. Lebowtiz. See webapage.

SuckerBets.txt
and
SuckerBetsAnalysis.txt,
to construct and analyze, respectively, *sucker's bets*, aka *non-transitive dice*
aka * Efron dice*.
They are accompanied by this article.

Capsules.txt
A Maple program to solve the Capsules puzzle published (starting Dec. 2016) in the New York Times magazine
and invented by Wei-Hwa Huang.

For example, the input file
produces, in 0.632 seconds, the output file

qFindRec.txt
A Maple program to guess linear recurrences with coefficients that are polynomials in q^{n} and q
for sequence of polynomials in q. (posted March 16, 2016).

FindRec.txt A Maple program to guess linear recurrences with polynomial coefficients. Written a long time ago, but posted March 16, 2016.

BiVariateMoms.txt A Maple program to automatically derive expressions for the average, variance, and higher moments of combinatorial random variables whose grand generating functions are rational in both variables. Read all about it in the article.

Pisot.txt A Maple program to investigate Pisot sequences. Read all about it in the article, by SBE, Neil Sloane, and DZ.

Skyscrapers.txt
A Maple program to solve (and generate!) Skyscrapers puzzles. It generated the following
puzzle book .

A435.txt A Maple program to investigate Neil Sloan'e "first love", OEIS sequence A435. Read all about it in the article .

Fqn.txt
A Maple program to investigate the Croot-Lev-Pach-Elenberg-Gijswijt "capset" breakthrough.
Read all about it in the article .

PERCY.txt
A Maple program to continue Major Percy A. Macmahon's heroic counting of tournamnets.
Read all about it in the article .

SVT.txt
A Maple program to study Singular Vector Tuples.
Read all about it in the article .

HaroldSilentShapiro.txt
A Maple program to study Rudin-Shapiro polynomials. It accompanies the
article about them.

GEPNER.txt
A Maple program to study Doron Gepner's intriguing permutation (and word) statistics, that we call "gep".
Read all about it in the article .

ALG.txt
A Maple package that decides whether a polynomial is the tensor product of lower-degree polynomials
It accompanies Manuel Kauers and Doron Zeilberger's
article.

[For the examples to work, you also need: AlgDataFile.txt]

CfiniteIntegral.txt a Maple package that finds recurrences and differential equations for integrals of powers of sequences of C-finite polynomials. It accompanies Shalosh B. Ekhad and Doron Zeilberger's article.

AFS.txt a Maple package that finds congruence theorems for a wide class of formal power series satisfying functional equations of a certain type, generalizing those studied by George Andrews, Aviezri Fraenkel, and James Sellers. It accompanies Shalosh B. Ekhad and Doron Zeilberger's article.

BFF.txt a Maple package that studies Disjoint Covering Systems It accompanies Shalosh B. Ekhad, Aviezri Fraenkel, and Doron Zeilberger's article.

CTcong.txt a Maple package that finds congruence theorems for certain partial sums of many combinatorial sequences. It accompanies William Y.C. Chen, Qing-Hu Hou and Doron Zeilberger's article.

stCore a Maple package that finds explicit expressions for the variance, and any desired finite moment (of course, the higher it is, the longer it takes) for the combinatorial random variable "SIZE of a simultaneous (s,t)-core partition", where s and t are coprime positive integers. It also does other things related to this statistic. It accompanies Shalosh B, Ekhad and Doron Zeilberger's interesting and deep article.

Sn
a Maple package that finds explicit expressions, and if not available,
recurrence relations, for sequences defined by sums of squares (and other powers)
of the characters of the symmetric group S_{n}, χ^{λ}(μ),
where the sums are over all shapes λ with a bounded number of rows, or included
in a meta-hook, and μ has `mostly' ones.
It accompanies Alon Regev, Amitai Regev,and Doron Zeilberger's
article.

[Added Oct. 23, 2015, a new version Sn.txt
contains a new procedure that lead to the following interesting
article.]

SBNYT, a Maple package to solve (and make) Spelling Bee type puzzles that started to appear in the New York Times magazine, Spring 2015. It requires the large data set milim, that should be downloaded to the same directory as the Maple package.

[For example, to get a puzzle-book with 100 puzzles, the input file yields, in a few seconds, the great puzzle book.

Gessel64 a Maple package that finds explicit polynomial expressions, in d, for the number of permutations of length d+r that do not contain an increasing sequence of length d, for symbolic d, and for numeric r. It also implements Ira Gessel's amazing determinant formula. It accompanies Shalosh B. Ekhad, Nathaniel Shar and Doron Zeilberger's article, dedicated to Ira Gessel on his 64th birthday.

CAcount a Maple package that implements a meta-algorithm that creats fast algorithms for computing the number of monomials left when a polynomial (with integer coefficients) is raised to high powers, and then taken mod 2 (and analogous things mod any prime). It accompanies Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger's article .

Four Maple packages accompany Shalosh B. Ekhad and Doron Zeilberger's article
The method(!) of Guess and Check:

W1D, for enumerating one-dimensional lattice
walks that never go negative,
W1Dp, for computing probabilities
of these walks,
W2D, for enumerating two-dimensional walks
that stay in the first quadrant, and
W3D, for enumerating three-dimensional walks
that stay in the first octant.

FunEq a Maple package that automatically proves (rigorously!) a wide class of functional equations that occur in enumerative combinatorics of maps, restriced walks, and other beasts. It accompanies Ira Gessel and Doron Zeilberger's article .

SLOANE75 and NEIL are two Maple packages that enuerates 123...d-avoiding words with the same number of occurrences of each letter. They are dedicated to Neil Sloane on his 75-th birthday, and accompany this article .

Words123 A Maple package that enuerates 123-avoiding words with the same number of occurrences of each letter. It accompanies Nathaniel Shar and Doron Zeilberger's article .

HIRSCH A Maple package that studies the Hirsch citation index (formerly "size of Durfee square") It accompanies Shalosh B. Ekhad and Doron Zeilberger's insightful article .

FOATA A Maple package that implements Dominique Foata's seminal bijection on permutations (and words), described in his article, Proc. Amer. Math. Soc. 19(1968), 236-240. It accompanies Doron Zeilberger's talk on Oct. 2, 2014, commemorating Foata's revolutionary work.

GOALS A Maple package for the automatic derivation of poynomial expression for the number of ways to have n teams (n=4 in the World Cup), all play each other, and each score r goals for, and r goals against. It accompanies the article, that also comes with the Maple package WorldCup that makes up neat puzzles.

GeometryMiracles A Maple package for the automatic discovery and proof of Plane Geometry Theorems regarding points and lines, and for enumerating successive generations. It is accompanied by the following article . That article also describes two additional Maple packages: RichardSerge, tat confirms the pleasant surprises of Richard Evan Schwartz and Serge Tabachnikov, and GeneralizedMorley, that discovers and proves generalizations of Morley's famous Trisector theorem.

NesApery A Maple package for the automatic discovery and proof of irrationaliy of interesting, and less interesting, `natural' constants, emulating Roger Apéry's seminal proof of the irrationality of ζ(3). It is accompanied by the following article .

AlgFunEq A Maple package for the automatic discovery and proof of explicit expressions for averages and higher moments of combinatorial statistics defined on Catalan families, with special emphasis on 132-avoiding permutations. It is accompanied by Shalosh B. Ekhad and Doron Zeilberger's article . A related minor Maple package is Cheyne, named after Cheyne Homberger, that handles statistics on 123-avoiding permutations.

MENAGES A Maple package for the automatic generation of enumeration theorems for generalized Ménage numbers. It is accompanied by Doron Zeilberger's article . A related minor Maple package is BALTIC.

TRIANGULATIONS written in collaboration with Alon Regev. A Maple package to study triangulations of an n-gon and its relation to other Catalan objects. [VERY PRELIMINARY, HOPEFULLY TO BE EXPANDED]

AutoSquared , A Maple package for the automatic generation of congruence automata for the fast computation of combinatorial sequences modulo m, for m not too big. It is accompanied by an article by Eric Rowland and Doron Zeilberger. In addition, there are the following specific Maple packages for the Catalan, Motzkin, and Delannoy numbers: CatalanLS, MotzkinLS, DelannoyLS.

qDYSON, A Maple package for computing ANY coefficient of the q-Dyson product, using the Karolyi-Nagy brilliant approach to the Zeilberger-Bressoud theorem. It is accompanied by an article by Shalosh B. Ekhad and Doron Zeilberger.

HIRSCHHORN, A Maple package for generating Ramanujan-style and Hirschhorn-style proofs of Ramanujan-type congruences. It is accompanied by an article by Edinah Gnang and Doron Zeilberger. The same article is also accompanied by another Maple package BOYLAN, that extends Matthew Boylan's listing in theorem 1.3 of his (Acta Arith. 111 (2004), 187-203) article.

AdinKingRoichman, A Maple package inspired by the
fascinating article by Ron Adin, Ronald King, and Yuval Roichman.
It was written during a visit of Ron Adin (May 6-10, 2013), and may one day be accompanied by an article joint with Ron Adin.

[Meanwhile the
input yields the
output

and the
input yields the
output
]

IPD, A Maple package for implementing the mini-revolutionary article by William Press and Freeman Dyson, to study the Prisoner's Dilemma.

GEA, A Maple package for generating Euler-Andrews style cautionary tales about the danger of naive empiricism, nevertheless using empirical methods! It is accompanied by an article by Shalosh B. Ekhad and Doron Zeilberger.

NesSomos, A Maple package for generating zillions of Somos-like miracles. It is accompanied by an article by Shalosh B. Ekhad and Doron Zeilberger.

ArithFormulas, A Maple package for enumerating, randomly generating, and optimizing zeroless representations of positive integers, only using 1 as inputs, and the binary operations plus, times, and power. It is accompanied by an article by Edinah K. Gnang and Doron Zeilberger.

TOWERS, A Maple package for the automatic enumeration of towers of k-mers, using the brilliant Bordelaise approach of Bétréma and Penaud. It is accompanied by an article.

SRINIVASA,
a Maple program that finds polynomial expressions
in terms of P,Q,R, of Phi_{r,s}(x) computed by hand
by Ramanujan in his article "On certain arithmetic functions",
Trans. Camb. Phil. Soc. v. 22, no. 9 (1916), 159-184.
It accompanies,
Doron Zeilberger's talk delivered on Nov. 7, 2012, at
Ramanujan 125 conference,
Univ. of Florida, Nov. 5-7, 2012, wonderfully organized by
Krishna Alladi and Frank Garvan.

For example, to see how Shalosh computed, in
less than half a second, the three tables
in the above article, the
input file, yields the
output file

To see many more entries, the
input file, yields the
output file

BinaryTrees, a Maple package that weight-enumerates, computes probability distributions, and does simulations with Binary trees. It accompanies an article by Larry Shepp, Doron Zeilberger, and Cun-Hui Zhang. The article is also accompanied by a drawing program called EtzBinary, and a preliminary, much smaller, program, ArbresBinaires.

GANAVIM,
a "trivial" Maple package that uses generatingfunctionlogy to conduct
statistical analysis of empirical data, something that is usually done by numeric packages.
It grew out of my disappointment at the
recent immoral policy of the Hillel Foundation to charge admission to their High Holiday services
from non-students.
(BTW, one can get much better prayers via the internet, see
this page.)
So far it does not accompany any article (and probably never will), but
to see a statistics of their exhorbitant "etnan" (a Hebrew word meaning *"prostitute's fee"*)

the input
yields
the output,

and since the data was already available, Shalosh also conducted
a statstical analysis of tuition costs, where

the input
yields
the output,

The aritcle by Brian Nakamura and Doron Zeilberger, about an efficient algorithm for enumerating generalized Wilf classes, is accompanied by the following Maple packages: P123, P1234, P12345, P123456, F123, F1234, and F12345.

BORDELAISE,
A Maple package that explores the amazing
École Bordelaise 3^{n} theorem and proof,
and its ramifications for random generation of domino towers, that I renamed *xaviers*

RandWilf,
A Maple package that uses Herb Wilf's seminal methodology for drawing,
uniformaly-at-random a member of a combinatorial set, applied to lattice walks
with any set of (non-negative) steps.
It accompanyies Doron Zeilberger's talk
Meta-GASCOMania, delivered at the GASCOM 2012 conference, Bordeaux, June 25-27,2012,
Tue. June 26, 2012, 9:00-10:00 (local time),
dedicated to the memory of Herbert Saul Wilf (1931-2011).

[For example, to see a random central Delauny path, the
input file
yields, almost immediately, the output file]

RITSUF, A Maple package that uses the C-finite ansatz to enumerate domino (and monomer-dimer) tilings of "skinny" plane regions. It is accompanied by an article by Shalosh B. Ekhad and Doron Zeilberger. Also mentioned there are the related Maple packages RITSUFwt and ARGF.

SolidSYT, A Maple package that enumerates Solid Standard Young Tableaux It is accompanied by an article by Shalosh B. Ekhad and Doron Zeilberger

mDimPars, A Maple package that computes polynomial expressions, in m, for the number of m-dimensional partitions of any specific, numeric, integer n (but as n goes beyond 12 it becomes very slow). It is accompanied by the article by Shalosh B. Ekhad.

GreeneNijenhuisWilf, accompanying the Doron Zeilberger's 119th opinion Farewell to "W" (Herbert Saul Wilf), a True VISIONARY for Whom EVERYTHING was INTERTWINED

DMP, In loving memory of my beloved guru Herb Wilf(1931-2012), that studies combinatorial objects that he talked about in problem number 6 of his Dec. 13, 2010 beautiful list of unsolved problems. DMP accompanies the article by Doron Zeilberger.

MARVIN, in fond memory of my dear friend Marvin Isadore KNOPP (1933-Dec. 24, 2011), to study Dedekind sums, and in particular to empirically confirm the Knopp idendity for these sums.

RPS, A Maple package that computes algebraic generating functions for enumerating words in any alphabet that have the property that they have the same number of occurrences, as consecutive substrings, of any two specified words of the same length. It is named after Richard Peter Stanley, who proposed a protypical problem. It is accompanied by the article by Shalosh B. Ekhad and Doron Zeilberger.

HIMURIM,
that tells you how to gamble optimally (in several senses),
if you are in a hurry.
It is accompanied
by the
article
by Shalosh B. Ekhad, Evangelos Georgiadis, and Doron Zeilberger.

(Another, smaller package
PURIM,
is also discussed there).

HANS Investigates the famous Rademacher Conjecture, and proves (in the usual sense of the word) that it is false. (the package was written in collaboration with Andrew V. Sills). It is accompanied by the article by Andrew V. Sills and Doron Zeilberger .

PARTITIONS automatically discovers and proves formulas for the number of (integer) partitions of n into at most m parts, for any numeric m, and lots of other things. (the package was written in collaboration with Andrew V. Sills). It is accompanied by the article by Andrew V. Sills and Doron Zeilberger .

Cfinite A C-finite Calculator and discoverer of new C-finite identities, and more importantly and impressively, factorizing C-finite sequences. It accompanies the article by Doron Zeilberger .

BallsInBoxes a Maple package for computing, both exactly and approximately, probabilities for the maximum number of balls co-habitating in the same box being such and such when n balls are thrown into m boxes. It accompanies the article by Shalosh B. Ekhad and Doron Zeilberger .

ChessWalks a Maple package for deriving generating functions for counting the number of walks any "chess piece" (even those that you make up) can walk on an a by b chessboard, with a and b finite, and staying on the board. It handles both the cases where it is supposed to return to its initial position, and the where it can end-up anywhere on the board. It generated the article by Shalosh B. Ekhad .

KamaShidukhim a Maple package for deriving generating functions for counting matchings in sequences of grid graphs, and much more general creatures. It accompanies the article by Shalosh B. Ekhad and Doron Zeilberger .

SameSexMarriages a Maple package for (semi-rigorously) proving the asymptotic normality of the number of same-sex marriages in a bisexual population. It accompanies the article by Shalosh B. Ekhad.

WalkPapers a Maple package for publishing articles about enueration of lattice walks (in any dimension) with any given set of steps, thar return to the starting point/ It accompanies the article by Shalosh B. Ekhad.

KamaEtzim a Maple package for deriving generating functions for counting the number of spanning tress of sequences of grid graphs, and much more general creatures. It accompanies the article by Shalosh B. Ekhad and Doron Zeilberger .

KamaTzviot a Maple package for deriving generating functions for chromatic polynomials of sequences of grid graphs, and much more general creatures. It accompanies the article by Shalosh B. Ekhad, Jocelyn Quaintance, and Doron Zeilberger .

ELIZALDE a Maple package for enumerating consecutive Wilf classes using the Umbral Transfer Matrix method and a generalization of the Goulden-Jackson method. It accompanies the article by Andrew Baxter, Brian Nakamura and Doron Zeilberger. Also accompanying this article the Maple package SERGI that does the same via the direct approach.

RookWalks a Maple package for counting multi-dimensional rook walks. It accompanies the article by Manuel Kauers and Doron Zeilberger.

LEON a Maple package for implementing some of the brilliant insights of Leon Ehrepreis (1930-2010). It accompanies the lecture given in Ehrenpreis' memory. [Added July 21, 2011: it also accompanies the article written in Leon's memory].

HISTABRUT a Maple package for symbol-crunching in probability theory, handling uni- and bi-variate distributions. It is accompanied by the article article by Doron Zeilberger.

DALIA a Maple package that implements the beautiful algorithm inplicit in Aviezri Fraenkel and Dalia Krieger's masterpiece The Structure of Complementary sets of integers: a 3-shift theorem, that appeared in Internat. J. Pure and Appl. Math. 10 (2004) 1--49. (Posted Aug. 4, 2010). It is accomopanied by a short description that also has sample input and output files.

HOOKER a Maple package that outputs recurrences and refined asymptotics for Berele-Regev (k,l)-hook-sums. It is accompanied by the article by Shalosh B. Ekhad and Amitai Regev .

GraphEnumeration a Maple package that counts unlabeled graphs and trees, following the excellent text Graphical Enumeration, by Frank Harary and Edgar Palmer. (Posted June 16, 2010, revised July 20, 2010). It is accomopanied by a short description that also has sample input and output files.

ISOMERS a Maple package that generates all isomers of a given compound by brute force. (Warning: very slow!) (First Posted June 16, 2010. Revised (with more detailed help), July 20, 2014 [Thanks to John Ogilvie]).

PEKERIS a Maple package that redoes 1958 Chaim Leib Pekeris' seminal calculations of the ground state energies of two electron atoms, on a 2010 computer. It accompanies an article by Christoph Koutschan and Doron Zeilberger (Posted May 26, 2010).

InvMaj a Maple package that studies the moments of the permutation statistics (Number of Inversions, Major Index) and their asymptotic distribution. It accompanies an article by Andrew Baxter and Doron Zeilberger (Posted April 7, 2010).

TZIUNIM a Maple package that computes letter grades for undergraduate classes, and automatically composes messages to students explaining their grades. (Posted Feb. 3, 2010).

JACK a Maple package that finds recurrences for Dyson-like constant terms in the style of I.J. Good. It is accompanied by the article A Eulogy for Jack Good (Posted Dec. 2, 2009).

BABUSHKAS a Maple package that counts the number of multi-set set-partitions, or more colorfully, the Number of Ways to reassmble several Russian Dolls (Posted Sept. 16, 2009).

MahonianStat a Maple package that studies and proves asymptotic normaility of the Mahonian distribution on words. It accompanies E. Rodney Canfield, Svante Janson, and Doron Zeilberger's article (Posted Aug. 14, 2009).

ENDRE, a Maple package that studies finite analogs of Szemeredi's theorem about arithmetic progressions. It accompanies Paul Raff and Doron Zeilberger's article (Posted July 13, 2009).

ChowRobbins,
a Maple package that tells you whether to **go** or to **stop**.
It accompanies Luis Medina and Doron Zeilberger's
article. That same article is also accompanied by Maple packages
STADJE,
computing probabilities of escape from "rational" half-planes in 2D lattice walks,
and WALKSab,
for finding the algebraic generating function for such walks (based on an algorithm
designed by Arvind Ayyer and Doron Zeilberger).
(Posted June 23, 2009).

MVPoisson, a Maple package that computes probabilities and expected values for certain important quantities in biochemistry. It accompanies Eduardo Sontag and Doron Zeilberger's article (Posted June 3, 2009).

LADAS, a Maple package that automatically generates, and then automatically proves, two-dimensional analogs of the 3x+1 conjecture. It accompanies Doron Zeilberger's article (Posted March 23, 2009). Another Maple package accompanying that article is COLLATZ, that only makes conjectures, but does not prove them.

PICAPIX, a Maple package that automatically generates, and solves, PicAPix puzzles. It generated Shalosh B. Ekhad's webbook One Hundred and Twenty Pic-A-Pix Problems (Posted Feb. 17, 2009).

CLT, a Maple package to compute asymptotics for moments, refining the Central Limit Theorem, and AsymptoticMoments, that finds asymptotic moments for discrete probability distributions that come up in enumeration. They are both accompanying an article By Doron Zeilberger . (Posted Dec. 4, 2008).

FPL, a Maple package to study the Razumov-Stroganov conjecture. It accompanies an article By Arvind Ayyer and Doron Zeilberger (Posted Dec. 3, 2008).

Ilse, Maple packages that incorporates a constant-term rendition of Ilse Fischer's gorgeous approach to counting Monotone Triangles. Very preliminary. It may accompany and/or inspire future papers by a non-empty subset of {DZ, Dan Romik, Ilse Fischer}. (Posted Oct. 6, 2008).

qTSPP, and TSPP, Maple packages that contain semi-rigorous proofs of George Andrews' and Dave Robbins' qTSPP conjecture, and of Stembridge's TSPP theorem, respectively. They accompany an article By Christoph Koutschan, Manuel Kauers and Doron Zeilberger (Posted Aug. 6, 2008).

AsyRec.txt, a Maple package that automatically computes the asymptotics for solutions of linear recurrence eqations with polynomial coefficients. It accompanies an article Doron Zeilberger. (First Posted April 2, 2008; Current version: April 26, 2016).

TANGLE, a Maple package that counts certain configurations that came up in studies of RNA. It accompanies an article By William Y.C. Chen, Jing Qin, Christian M. Reidys and Doron Zeilberger (Posted Feb. 23, 2008).

BruteTwoFone, a Maple package that searches, by (clever!) brute force, for hypergeometric closed-form "strange" evaluations. It accompanies Moa Apagodu and Doron Zeilberger's article (Posted Feb. 22, 2008).

ConnectFour, a Maple package that automatically generates Connect Four End-Game Problems. It generated Shalosh B. Ekhad's webbook One Hundred and Twenty Connect-Four End-Game Problems (Posted Feb. 1, 2008).

DRUNKARD,
a Maple package that computes the probabilities of return to
the origin of a simple random walker in Z^{d} that
starts at the origin, and is either unconfined
(like in Polya's original case), or must stay in
(a) the positive orthant (b) the simplex
x_{1} ≥ x_{2} ≥ ... ≥ x_{d}
(c)x_{1} ≥ x_{2} ≥ ... ≥ x_{d} ≥ 0,
as well as counting number of ways of walking from A to B etc.
It accompanies Doron Zeilberger's
article (Posted Dec. 19, 2007).

LUACH, a Maple package that converts Hebrew dates to Gregorian ones, and vice versa, and also constructs Hebrew Yearly Calendars, for ANY year, after the creation of the world, for ever after, and has other neat features like predicting the year when Rosh Hashana will be at any given Gregorian date, for example Christmass (answer: 20606). It accompanies Doron Zeilberger's lecture, at the Rutgers University Experimental Mathematics Seminar, on 5:00pm, Tishrei 8, 5768, entitled: "Why Erev Yom Kippur can never be on Thursday Evening". (Posted Oct. 7, 2007).

RotaStanley, a Maple package that uses Gian-Carlo Rota's Umbral Calculus to enumerate Richard Stanley's P-Partitions. It accompanies Shalosh B. Ekhad and Doron Zeilberger's article (Posted June 12, 2007).

Bipartite, a Maple package that counts regular (labelled) bipartite graphs It accompanies Shalosh B. Ekhad and Doron Zeilberger's article In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses? Also accompanying this article is the Maple package LatinRectangles, for computing the number of Latin Rectangles with small width but with rather long length. (Posted Jan. 2, 2007).

VATTER, a Maple package that finds Generalized Enumeration Schemes (in the sense of Vince Vatter) for Wilf classes. It accompanies Doron Zeilberger's article On Vince Vatter's Brilliant Extension of Doron Zeilberger's Enumeration Schemes for Herb Wilf's Classes (Posted Jan. 2, 2007).

AMITAI, a Maple package that makes conjectures about the number of n-celled Young tableaux with certain properties. It accompanies Doron Zeilberger's article Proof of a Conjecture of Amitai Regev about Three-Rowed Young Tableaux (and Much More!) (Posted Dec. 11, 2006).

FELLER, a Maple package that studies Coin Tossing, a la Feller's v. 1 ch. 3. It accompanies Doron Zeilberger's article Fully Automated Computerized Redux of Feller's (v.1) Ch. III (and Much More!) (Posted Nov. 14, 2006).

AppsWZ, a Maple package that uses the Almkvist-Zeilbeger algorithm to compute recurrences for quantities of interest in enumeration and discrete probability and AppsWZmulti, that does the same for multi-dimensional lattice paths and random walks. They accompany Moa Apagodu and Doron Zeilberger's article FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability (Posted Oct. 20, 2006).

RUIN, a Maple package that studies the Gambler's Ruin problem. It accompanies Doron Zeilberger's article Symbol Crunching with the Gambler's Ruin Problem (Posted Sept. 17, 2006).

EMILIE,
a Maple package whose first version empirically verifies and conjecutures
a general generating function for Paul Heideman and Emilie Hogan's
beautiful article
New Family of Somos-like Recurrences
. The current version completes the lovely proof of
Heideman and Hogan by doing the "base cases" (completely empirically,
yet rigorously! It is an amazing illustartion of experimental-yet-rigorous
math). If you type EmiliePaul(K,n); while in
EMILIE, you would get
oEmiliePaul, that discovers everything from scracth!
(i.e. it didn't save any previously computed results).

(First version Posted March 28, 2006. This version: Sept. 19, 2006).

DET, a Maple package that automatically guesses(!) and proves(!!) explicit determinant evaluations of matrices with holonomic (in particular, rational-function) entries. It accompanies the article The Holonomic Ansatz II. Aoutomatic Discovery(!) and Proof(!!) of Holonomic Determinant Evaluations (Posted Feb. 20, 2006).

GuessHolo2, and

GuessHolo3 are two Maple packages that guess holonomic
representations for discrete functions of two and three variables
respectively, and apply them for conjecturing (and proving!)
pure reecurrence relations for discrete functions enumerating
lattice paths.
They accompany by the article
The Holonomic Ansatz I. Foundations and Applications
to Lattice Path Counting
(Posted Feb. 5, 2006).

TILINGS, a Maple package that automatically computes generating functions enumerating tilings of rectangular boards of arbitrary but fixed width (but general length), by the tiles belonging to any inputted set of tiles. It accompanies by the article Automatic CounTilings (Posted Jan. 20, 2006).

PHIL,
a Maple package that implements (and empirically verifies!)
Phlip Matchett Wood's lovely article
"A biective proof of

f_{n+4}+1f_{1}+2f_{2}+ ... +
nf_{n}=(n+1)f_{n+2}+3"

(Posted Dec. 9, 2005).

SMCramsey, a Maple package that computes (symbolic!) moment for random variables that arise in Graph Colorings and Ramsey theory. It accompanies by the article Symbolic Moment Calculus II.: Why is Ramsey Theory Soooo Eeeeenormously Hard? (Posted Nov. 22, 2005).

Bearoff, BearoffOneDie, Sulam, are three Maple packages that study the bearoff stage in Backgammon (generalized for an arbitary fair die). They are accompanied by the article How to Play Backgammon (if you must), and how to Research it (if you have time) (Posted Nov. 15, 2005).

JONAS, a Maple package that implements the lovely bijective proof of Jonas Sjostrand . It is accompanied by the article Another Proof that Euler Missed: Jonas Sjöstrand's Amazingly Simple (and Lovely!) Proof of the No-Longer-So-Amazing Lattice Paths Conjecture (Posted Oct. 23, 2005).

TEN, a Maple package that automatically proved the amazing Loehr-Warrington 10 to the power n conjecture. It is accompanied by the article A Proof of the Loehr-Warrington Amazing TEN to the Power n Conjecture (Posted Sept. 14, 2005).

SuDoku, a Maple program to solve SuDoku problems.
(Posted July 20, 2005).

For example, the
input file produces the
output file.

Added Feb. 16, 2006:
Bobby Griffin just won ten dollars in my
Experimental Math Class contest
for the best Maple Sudoku program
SuDokuGriffin.
It is a thousand times faster!

To prove my point, the
input file produces the
output file.

Five Maple packages accompany Multi-Variate Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory by M. Mohammed and D. Zeilberger. Namely:

- MultiZeilberger, for hypergeometric multiple summation,
- MultiZeilbergerDen, for hypergeometric multiple summation with denominators,
- qMultiZeilberger, for q-hypergeometric multiple summation
- MultiAlmkvistZeilberger, for hypergeometric multiple intgeration.
- SMAZ, for hypergeometric multiple intgeration of symmetric integrands.

QuantumMACMAHON, a Maple package that verifies the Quantum Macmahon Master Theorem for any specific dimension. It accompanies the (still incomplete) article The Quantum MacMahon Master Theorem by S. Garoufalidis, T. Le and D. Zeilberger.

(WIth Andrew V. Sills) GuessRat, a Maple package that guesses Rational function for empirically derived data; and GoodDyson, taylored-made for Dyson-type constant-term identities. Both accompany Disturbing the Dyson Conjecture (in a GOOD way) by Sills and Zeilberger.

ZEILBERGER, and qZEILBERGER, two Maple packages that perform the simplified Zeilberger and q-Zeilberger algorithms as described in the article Sharp Upper Bounds for the Orders of the Recurrences Outputted by the Zeilberger and q-Zeilberger Algorithms by Mohamud Mohammed and Doron Zeilberger. (posted Aug. 10, 2004).

MarkovWZ,
A Maple package (written in collaboration with Mohamud Mohammed)
that computes Markov-WZ pairs and MWZ Acceleration
schemes.
It accompanies the article
The Markov-WZ Method
by D. Zeilberger and M. Mohammed..
Also accompanying this paper are the Maple packages
MarkovAZ, for computing Markov-Almkvist-Zeilberger pairs and
ContMarkovWZ, for computing Continuous-Discrete Markov-WZ pairs.
(posted June 3, 2004).

Added July 19, 2004: MarkovWZ has been upgraded to do diagonal
contours, download the enhanced package
MarkovWZdiag, and type ezraD(); for specifics.

CheckMate.txt, a Maple package for solving Mate-in-k-Moves,
Very preliminary and very primitive, hopefully to be improved
in the future. You should also download, and keep in
the same directory, the files
MateInThree.txt [last update: Jan. , 2011] and
MateInTwo.txt
[last update: March 31, 2018],
(First Posted June 18, 2004)..

For 253 Mate-In-Two problems see this
problem book ,
and here the problems with their
solutions

TAG, a Maple package for studying the Totally Ambiguous Grammar discussed in Doron Zeilberger's talk at the Princeton Discrete Math Seminar (March 31, 2004). It may accompany a forthcoming article by Doron Zeilberger.

SMCper, A Maple package that computes Pattern Statistics for permutations. It accompanies the article Symbolic Moment Calculus I.: Foundations and Permutation Pattern Statistics .

BYRNES, A Maple package that investigates Chomp. It accompanies the article Chomp, Recurrences, and Chaos(?). You should also download the data file ChompData, and put them in the same directory.

GRW, A Maple package to handle the Gelfand-Retakh-Wilson "non-commutative" theory of "quasi-determinants". [Still Very Preliminary, Version of Feb. 21, 2003].

OtiotUmisparim, [Last Update, Aug. 3, 2003, with new procedure PtorKefWm, that solves multiplication Alphametics with one typo, like the one in Yediot, Aug. 1, 2003]. A Maple package that solves, and more impressively generates, `alphametics' (cryptharithms in which every number is a word). It was used to generate Shalosh B. Ekhad's web-book Computer-Generated Alphametics.

[Added June 28, 2015: Another type of puzzle that sometimes appears in Yediot, and puzzle magazines, has a long multiplication problem where all the occurrences of a certain digit is marked X, and the remaining ones are left blank. This type of (annoying!) puzzles is handled in the Maple package GILAD.txt].

[Added Aug. 13, 2016: A much faster version (using pre-computed date, that only works for the usual 3-digit by 3-digit multiplication
problems (in base 10) is the Maple package
GILADfast.txt.

Here
is a list (called G33) of 49093 problems and solutions. In fact all the problems of
3-digit by 3-digit multiplication using exactly 9 digits (i.e. exactly one is missing) and having
a unique solution.

Added Aug. 21, 2016: Try out any of the 200 challenging problem in this computer-generated
puzzle book
].

CLD, A Maple package that accompanies the article Lieber Opa Paul , Ich Bin Auch Ein Experimental Scientist, that automatically discovers, and proves at the same time, explicit evaluations of Hankel and Toeplitz determinants, using the Dodgson ansatz.

EHRENBORG, A Maple package that accompanies the article I Am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures. But One is FAMOUS, While The Other Is FALSE, by Doron Zeilberger, (placed here July 2, 2002). This package also contains procedures that implement the Robinson-Schenstead Correspondence, and its inverse, RS and iRS, respectively, (see the on-line help (ezra();)), that may be of independent interest.

There are five Maple packages accompanying the article Computerized Deconstruction, by Doron Zeilberger. These are (in order of appearance in the paper): PADE, AperyWZ, AperyRecurrence, AperyAcc, and AperyAppx. Look at the webpage of the article for links to sample input and output files.

AARON, A small Maple package that accompanies the article Refined Restricted Permutations, by Aaron Robertson, Dan Saracino, and Doron Zeilberger. It requires WILF, see below. Download it as AARON, get into Maple, and type ez(); for on-line help, not to be confused with ezra(); that describes the parent package WILF (see below).

LinDiophantus.txt,
A Maple analog of the Andrews-Paule-Riese
OMEGA package for Macmahon's method for solving linear
diophantine equations. Unlike the A-P-R treatment that resorts
to dubious analysis, my approach is completely elementary (high-school
algebra), and hence more rigorous.

Added Sept. 19, 2012: recently I added some procedures to compute
generating functions for the Condorcet so-called paradox.

Look at sample input and output files
here.

There are three Maple packages accompanying my article The Umbral Transfer-Matrix Theorem V. The Goulden-Jackson Cluster Method for Infinitely Many Mistakes. These are UGJ, for the general case, SymUGJ, for the case where the mistakes are invariant under the action of the Symmetric Group, and SiPerUGJ, for the case where the mistakes are invariant under the action of the group of Signed Permutations. To use, for example, UGJ, first download it as UGJ, go into Maple, by typing maple (ENTER), then type : read UGJ: (ENTER) and then follow the on-line instructions. Similarly for SymUGJ and SiPerUGJ.

There are three Maple packages accompanying my article The Umbral Transfer-Matrix Theorem IV. Counting Self-Avoding Polygons and Walks. The first one is USAP, A Maple package for computing and using Umbral Schemes for important families of self-avoiding polygons. The second one is USAW, A Maple package for computing and using Umbral Schemes for important families of self-avoiding walks. The third one is MAYLIS, A Maple package for handling, Umbrally, various classes of convex polyominoes by perimeter. To use, for example, USAP, first download it as USAP, go into Maple, by typing maple (ENTER), then type : read USAP: (ENTER) and then follow the on-line instructions.

ZOO, A Maple package for computing and using Umbral Schemes for important families of lattice animals. It accompanies my paper on that subject. (Placed here: Dec. 27, 2000).

PERCY, A Maple package for handling Markovian Permutation Statistics It accompanies Foata and Zeilberger's paper on that subject. (Placed here: Oct. 31, 2000).

PPar, A Maple package for computing and using Umbral Schemes for plane partitions and monotone triangles. It accompanies my paper on that subject. (Placed here: Oct. 26, 2000).

PARRONDO, A Maple package for studying the Parrondo Paradox. It accompanies Ekhad and Zeilberger's article Remarks On The PARRONDO PARADOX.

Chomp3Rows, A Maple package for finding symbolic algorithms for playing 3-Rowed Chomp. It accompanies my article Three-Rowed CHOMP.

PEG, A Maple package for solving Peg-Solitaire on arbitrary boards, including the standard `English' one and triangular ones (Hi-Q). One day it might lead to an article, but right now it is very preliminary. First posted: May 12, 2000. Last update: May 12, 2000.

ROTA, A Maple package for handling Umbral Schemes. It accompanies my paper on that subject.

There are seven Maple packages described in Maple Packages For Counting Skinny Physical Creatures. (http://sites.math.rutgers.edu/~zeilberg/tm.html) It accompanies my paper on that subject.

CGJ , A Maple package that accompanies Anne Edlin and Doron Zeilberger's paper The Goulden-Jackson Cluster Method For Cyclic Words. It generates generating functions counting necklaces that do not contain any ugly segments.

RORTY , A Maple package that accompanies Ekhad and Zeilberger's paper A User's Manual for RORTY: A Maple Package for Finding Final Vocabularies..

JAN , A Maple package that accompanies Ekhad and Zeilberger's paper There Are More Than 2**(n/17) n-Lettered Ternary Words. To use JAN, download it as JAN go into Maple, type `read JAN`, and follow the instructions given there.

WILF , and
HERB ,
Two Maple packages that automatically find
(rigorously and empirically, respectively) Enumeration Schemes
for counting permutations with forbidden patterns
(Wilf classes).
They accompanies
Doron Zeilberger's paper
Enumeration Schemes, and More Importantly,
Their Automatic Generation.
To use WILF, download it as WILF
go into Maple, type `read WILF`, and follow the
instructions given there. (ditto for HERB.)

Added July 19, 2004: A very significant enhancement of WILF
has been just completed by my brilliant student
Vince Vatter, in his Maple package
WILFPLUS. It does many more cases than WILF, but in a few
cases runs a bit slower, so I am still keeping WILF here.

RENE.txt ,
Last Update: Feb. 6, 2012
[Monthly problem 11615 added, courtesy of Matthew Russell].
*Tested on Maple9 and below.*
A Maple package that gives
very elegant and succinct Maple-statements of
fifty five famous theorems in Plane Geometry,
and 14 Monthly Problems, that
enable their immediate proof by Maple. It makes explicit the
350-year-old secret, still kept away from school-children, that
Plane Geometry is all routineable. This Maple package was adapted
(and largely extended) from
Shalosh B. Ekhad, XIV's Elementary Geometry Textbook(ca. 2050) .

RON , A Maple package to study Ron Graham's problem on Schur triples. It accompanies Aaron Robertson and Doron Zeilberger's paper A 2-Coloring of [1,N] Can Have (1/22)N^2+O(N)... . To use it, download it as RON go into Maple, type `read RON:`, and follow the instructions given there.

DODGSON , DODGSON: VERY PRELIMINARY version of a package that uses Dodgson rule to conjecture `closed form' expressions for (so far Hankel) determinants. Soon it will prove the conjectured expression, etc. etc. Written By Tewodros Amdeberhan and Doron Zeilberger. Accompanying a forthcoming paper by them. To use it, download it as DODGSON go into Maple, type `read DODGSON:`, and follow the instructions given there.

IRRAT , Investigates how Apery could have arrived empirically at his marvelous irrationality proofs. VERY PRELIMINARY

HORTON. A Maple package to study John HORTON Conway's brilliant Audiactive Decay. It accompanies Shalsoh B. Ekhad and Doron Zeilberger's paper on that subject .

John Noonan and I have written several packages dealing with the Goulden-Jacskon Cluster method. It accompanies our paper on that subject.

LEGO, A Maple package that automatically counts (certain) LEGO towers. This package is a companion to the paper Automated Counting of LEGO Towers.

DIKDUK, A Maple package that attemps to derive the (so-far Markovian) grammar of an inputted language (given in terms of a finite set of words), and then computes the generating function of the (infinite) language whose grammar it is. Very preliminary version! Better versions coming up. To use it, download it as DIKDUK, go into Maple, type `read DIKDUK:`, and follow the instructions given there .

SAWgrammar, A Maple package that uses the above package DIKDUK to find generating functions for families of restricted Self Avoiding Walks in a strip and variations. You must have DIKDUK at the same directory. To use it, download it as SAW, go into Maple, type `read SAW:`, and follow the instructions given there .

DREIDEL, A Maple package that plays the dreidel for any number of players, and computes winning probabilities, and expected duration, for 2-player games. To use it, download it as DREIDEL, go into Maple, type `read DREIDEL:`, and follow the instructions given there .

DOMINO, a Maple package that studies the dimer problem. To use it, download it as DOMINO, go into Maple, type `read DOMINO:`, and follow the instructions given there. [Last Update: Oct. 10, 1997]

LUC , A Maple program that implements the amazing explicit formula for the Jack polynomials, found by Luc Lapointe and Luc Vinet, in their paper ``A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture''. To use it, download it as LUC, go into Maple, type `read LUC:`, and follow the instructions given there.

ROBBINS , a Maple package accompanying my paper ``Proof of the alternating sign matrix conjecture ( Electronic Journal of Combinatorics. 3(2)(1996) [Foata issue] R13). It empirically verifies every non-trivial statement in the (structured) proof. To use it, download it as ROBBINS, go into Maple, type `read ROBBINS:`, and follow the instructions given there.

SCHUTZENBERGER.txt, a Maple package that treats algebraic and other formal power series. To use it, download it as SCHUTZENBERGER.txt, go into Maple, type `read SCHUTZENBERGER.txt:`, and follow the instructions given there.

SYLVESTER , A Maple implementation of Sylvester's enhancement of Tchebychev's method for finding lower and upper bounds for psi(x)/x, based on his CW IV, paper 72. To use it, download it as SYLVESTER go into Maple, type `read SYLVESTER:`, and follow the instructions given there.

You can also find:

MIKLOS, a short program/article, by AAron Robertson, Herb Wilf, and Doron Zeilberger, to find generating functions for permutations with NO (resp. exactly ONE) occurences of the pattern 132 and exactly r occurences of the pattern 123. Save this program as MIKLOS, go into Maple, type: read MIKLOS; and follow the instructions.

LOU, A Maple package that investigates Lou Kauffman's approach to 4CT. It accompanies the article by Bobbe Cooper, Eric Rowland and Doron Zeilberger. .

NSFdms, A Maple package for doing NSF-research in Mathematics, or rather, meta-mathematical research, about the funding of the Division of Mathematical Sciences of the National Science Foundation. It should be used in conjuction with DMSDATA, (download this file in the same directory). To find out how to use it, download User's Manual for NSFdms. Written: Oct. 6, 1998

GetMicrosoftJob, A Maple package that solves the Microsoft job-interview puzzle, about the U2 band crossing a bridge at night. To use GetMicrosoftJob, download it as GetMicrosoftJob, go into Maple, type `read GetMicrosoftJob;`, and follow the instructions given there. Written: Sept. 25, 1998

ChutesAndLadders, A Maple package that plays Milton-Bradley's ChutesAndLadders(tm) game, and generalizations with arbitrary sets of Chutes and Ladders, and computes probabilities of winning, and expected duration of the game. To use ChutesAndLadders download it as ChutesAndLadders go into Maple, type `read ChutesAndLadders`, and follow the instructions given there. Written by Doron Zeilberger, with Hadas Zeilberger (b. Dec. 19, 1990) serving as consultant.

RubikClock , A Maple program to solve Rubik's Clock Puzzle. To use it, download it as RubikClock, go into maple, and type: read RubikClock: Then follow the on-line instructions.

BET , A short Maple package that studies the Ben Ezra Transform, as it is described in Ibn Ezra's Sefer Hamispar, Gate 6.

BenEzra, implements (the general) Rabbi Ben Ezra (Ibn Ezra) pseudo-random drowning of bad guys (essentially the Josephus problem).

rec2 , Empirically finds linear recurrences, in each variables, with polynomial coeffs. satisfied by a discrete function a(n,m).

appel, A small Maple package that implements Andrew Appel's Test for mathematical content. It accompanies Shalosh B. Ekhad paper The Theory of Self-Avoiding Walks in only 66.81 percents mathematical (according to the Appel test).

pinsky, A small Maple package that answers a Calculus excercise posed by Mark Pinsky.

alice, A small Maple package accompanying D. Zeilberger's paper Dodgson's Determinant Evaluation Rule Proved by TWO-TIMING MEN and WOMEN.

rabbit , A small Maple package accompanying T. Amdeberhan and S.B. Ekhad's paper A Condensed Condensation Proof of a determinant evaluation conjectured by Greg Kuperberg and Jim Propp.

catalan , A small Maple package accompanying my paper (exclusive for my http and ftp archives) ``Catalan strikes again (and again*)''

hex , A short Maple program that proves Pascal's hexagon's theorem, that only works for Maple 5 and below (Note: I thank Tian Bo for pointing it out. Please use procedure Pascal() from RENE )

change.txt , How many ways can you make change?

elim , Naive elimination in the ring of linear partial difference operators with polynomial coefficients