arXiv subjects
Publications and source records attributed to Shalosh B. Ekhad.
Yet another example where "physical" (i.e. only checking finitely many special cases) gives a fully rigorous proof, notwithstanding what your "Intro To Proofs" prof told you!
Richard Stanely proposed, in a recent Amer. Math. Monthly Problem, to prove a nice explicit formula for the generating function for the number of n-letter words in {H,T} that have as many occurrences of HT as HH. In this article, we show how to prove this problem automatically, and ANY problem of that type, regardless of the size of the alphabet and the length of the two chosen strings
Binomial Theorem for (N+n)^r is described with non-commuting variables N and n.
Why bother with fully rigorous proofs when one can very quickly get semi-rigorous ones? Yes, yes, we know how to get a "rigorous" proof of the result stated in the title of this article. One way is the boring, human one, citing some heavy guns of theorems that already exist in the literature. We also know how to get a fully rigorous proof automatically, using the methods in this http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.htm neat article (but it would be a little more complicated, since the probability generating polynomial is not "closed form" but satisfies a second-order recurrence gotten from the Zeilberger algorithm), otherwise the same method would work, alas, it is not yet implemented. Instead, we chose to use the great Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/HISTABRUT">HISTABRUT(in fact, a very tiny part of it, procedure AlphaSeq), explained in this other http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/histabrut.html">neat article, and get a semi-rigorous proof. We also needed the nice little Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/GuessRat">GuessRat, to do the guessing of rational functions. Equipped with these two packages, Zeilberger wrote a short Maple program http://www.math.rutgers.edu/~zeilberg/tokhniot/SameSexMarriages">SameSexMarriages that enabled the author to generate this paper.
We comment on, elaborate, and extend the work of Warren Ewens and Herbert Wilf, described in their http://www.pnas.org/content/104/27/11189.full.pdf about the maximum in balls-and-boxes problem. In particular we meta-apply their ingenious method to show that it is not really needed, and that one is better off using the so-called Poisson Approximation, at least in applications to the real world, because extremely unlikely events mever happen in real life. This article is accompanied by the Maple package http://www.math.rutgers.edu/~zeilberg/tokhniot/BallsInBoxes">BallsInBoxes.
This short article, dedicated to our beloved guru Philippe FLAJOLET (1948-2011), is a case-study in computer-generated combinatorial research, where the computer, all by itself, is using the transfer-matrix method to derive (rigorously!) rational generating functions for chromatic polynomials for infinite sequences of graphs generalizing the action of taking the Cartesian product with a path of length n, n=1,2,... .
This short article answers a question asked by Bruce Sagan in the last few minutes of his wonderful talk (http://www.youtube.com/watch?v=Fdn890jg2U0), (describing joint work with Carla Savage) at the Rutgers University Experimental Mathematics seminar delivered on Dec. 9, 2010.
This is an etude in experimental semi-rigorous (rigorizable!) mathematics. The leading asymptotics was brilliantly derived by Allan Berele and Amitai Regev for general hooks H(k,l) and general powers z, but what about more refined asymptotics? For small k and l, one can "guess" a linear recurrence (since we live in the holonomic ansatz) and using the Birkhoff-Trjitzinsky method, beautifully implemented in Doron Zeilberger's Maple package AsyRec (that has been incorporated into the present Maple package), we computed amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev asymptotic formula, and especially the impressive constant in front!
Richard Stanley proved that the centralized/normalized version of the random variable "length of largest up-down subsequence" in a random permutation of length n is asymptotically normal. We go beyond and present a more refined asymptotic formula for the moments.
We prove, via 30 seconds of Maple computation, that there are 10^n words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be immediately followed by a -2.
Here we present the reasoning behind, and program to find, the generating functions for the number of permutations in the title. The article duals as the "accompanying" Maple package.
In this reverse archeological find, we see how mathematics will be written in fifty years.
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for which no extant proof existed, is given a computer-generated proof, hopefully for good.
George Andrews's recent challenge to automated identity-proving and the WZ method is dealt with. It is argued that the rivalry between the classical and automated approaches to hypergeometric sums is beneficial to both.
A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.
We give a generalization and a short mechanized proof of determinant conjectured by G. Kuperberg and J. Propp. Further generalizations and applications of the method to some q-analogues may be found in http://www.math.temple.edu/~tewodros
We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov[KM].
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.