Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations. In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find that they can always be decomposed into so-called 'atomic' permutations introduced in this work. This decomposition allows us to enumerate the (circular) permutations of a subset of $\mathbb{N}$ by the length of their runs. Furthermore, we rederive, in an elementary way and using the methods developed here, a result due to Kitaev on the enumeration of valleys.
@article{10_37236_4235,
author = {Christopher J. Fewster and Daniel Siemssen},
title = {Enumerating permutations by their run structure},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/4235},
zbl = {1298.05010},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4235/}
}
TY - JOUR
AU - Christopher J. Fewster
AU - Daniel Siemssen
TI - Enumerating permutations by their run structure
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/4235/
DO - 10.37236/4235
ID - 10_37236_4235
ER -
%0 Journal Article
%A Christopher J. Fewster
%A Daniel Siemssen
%T Enumerating permutations by their run structure
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/4235/
%R 10.37236/4235
%F 10_37236_4235
Christopher J. Fewster; Daniel Siemssen. Enumerating permutations by their run structure. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/4235
