In this paper we study, given a group $G$ of permutations of a finite set, the so-called fixed point polynomial $\sum_{i=0}^{n}f_{i}x^{i}$, where $f_{i}$ is the number of permutations in $G$ which have exactly $i$ fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014.
@article{10_37236_2955,
author = {C. M. Harden and D. B. Penman},
title = {Fixed point polynomials of permutation groups.},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {2},
doi = {10.37236/2955},
zbl = {1280.20001},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2955/}
}
TY - JOUR
AU - C. M. Harden
AU - D. B. Penman
TI - Fixed point polynomials of permutation groups.
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/2955/
DO - 10.37236/2955
ID - 10_37236_2955
ER -
%0 Journal Article
%A C. M. Harden
%A D. B. Penman
%T Fixed point polynomials of permutation groups.
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/2955/
%R 10.37236/2955
%F 10_37236_2955
C. M. Harden; D. B. Penman. Fixed point polynomials of permutation groups.. The electronic journal of combinatorics, Tome 20 (2013) no. 2. doi: 10.37236/2955
