Zeons, Permanents, the Johnson scheme, and Generalized Derangements

TL;DR

This paper explores the connections between zeon algebra, permanents, Johnson schemes, and generalized derangements, introducing new formulas and relationships in algebraic combinatorics.

Contribution

It introduces a permanent trace formula analogous to MacMahon's Master Theorem and links permanents to Johnson schemes and generalized derangements.

Findings

Derived moment polynomials related to the exponential distribution.

Established a permanent trace formula similar to MacMahon's Master Theorem.

Connected matrix entries to variations of derangements and permutation group actions.

Abstract

Starting with the zero-square "zeon algebra" the connection with permanents is shown. Permanents of sub-matrices of a linear combination of the identity matrix and all-ones matrix leads 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.