arXiv subjects
Publications and source records attributed to Doron Zeilberger.
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.
Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain $n(n-1)/2$-dimensional polytope is given by the product of the first n-1 Catalan numbers.
The impact of the computer on present and especially future mathematics is illustrated by means of the iconic example of WZ theory.
We prove that the `connective constant' for ternary square-free words is at least $2^{1/17} = 1.0416 ... $, improving on Brinkhuis and Brandenburg's lower bounds of $2^{1/24}=1.0293 ...$ and $2^{1/22}=1.032 ...$ respectively. This is the first improvement since 1983.
Rabbi Abraham Ben Meir Ibn Ezra's(1089-1164) computation of the seventh row of the so-called Pascal-Chu triangle is described.
In this reverse archeological find, we see how mathematics will be written in fifty years.
David Brydges and Thomas Spencer's Lace Expansion is abstracted, and it is shown how it sometimes gives rise to sieves.
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for which no extant proof existed, is given a computer-generated proof, hopefully for good.
A combinatorial problem that came up in Combinatorial Chemistry is solved.
The Rev. Dodgson's determinant condensation rule is given a bijective proof.
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns `abc',`cab' and `abcd'.
We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1,n] can have is (n^2)/22 + O(n). This was solved independently by Tomasz Schoen.
The celebrated Foata combinatorial model for Hermite polynomials, and his seminal and beautiful proof of the Mehler formula, are straightened to deal with two sexes rather than one, with the exclusion of same-sex relationships (both marital and non-marital).
The powerful (and so far under-utilized) Goulden-Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of `dirty words', is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website (http://www.math.temple.edu/~zeilberg/gj.html), are described and explained.
We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.
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.
Catalan's formula, for the portion of the inheritance that a legitimate child of a 19th-century deceased French gentleman should receive, is given a new proof (using Difference Operators), and generalized. Another, more computationally efficient, formula is also derived.