3,385 research outputs found
Automatic Enumeration of Generalized Menage Numbers
I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook
polynomials and the so-called C-finite ansatz, fully implemented in the
accompanying Maple package
(http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES, that
reproduces in a few seconds, rigorously-proved enumeration theorems on
permutations with restricted positions, previously proved by quite a few
illustrious human mathematicians, and that can go far beyond any human
attempts.Comment: 15 pages. An extended version of the last of three invited talks
given by the author at the 71th Seminaire Lotharingien de Combinatoire, that
took place in Bertinoro, Italy, Sept. 16-18, 201
Zero-sum problems for abelian p-groups and covers of the integers by residue classes
Zero-sum problems for abelian groups and covers of the integers by residue
classes, are two different active topics initiated by P. Erdos more than 40
years ago and investigated by many researchers separately since then. In an
earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60],
the author claimed some surprising connections among these seemingly unrelated
fascinating areas. In this paper we establish further connections between
zero-sum problems for abelian p-groups and covers of the integers. For example,
we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If
{a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly
2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there
exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in
I}c_s=0. Our main theorem in this paper unifies many results in the two realms
and also implies an extension of the Alon-Friedland-Kalai result on regular
subgraphs
History of Catalan numbers
We give a brief history of Catalan numbers, from their first discovery in the
18th century to modern times. This note will appear as an appendix in Richard
Stanley's forthcoming book on Catalan numbers.Comment: 10 page
- …