International Journal of Combinatorics
Volume 2011 (2011), Article ID 539030, 29 pages
doi:10.1155/2011/539030
Research Article

Zeons, Permanents, the Johnson Scheme, and Generalized Derangements

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA

Received 20 January 2011; Accepted 1 April 2011

Academic Editor: Alois Panholzer

Copyright © 2011 Philip Feinsilver and John McSorley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Starting with the zero-square “zeon algebra,” the connection with permanents is shown. Permanents of submatrices of a linear combination of the identity matrix and all-ones matrix lead to moment polynomials with respect to the exponential distribution. A permanent trace formula analogous to MacMahon's master theorem is presented and applied. Connections with permutation groups acting on sets and the Johnson association scheme arise. The families of numbers appearing as matrix entries turn out to be related to interesting variations on derangements. These generalized derangements are considered in detail as an illustration of the theory.