In this paper, we establish a natural bijection between the almost-increasing cyclic permutations of length $n$ and unimodal permutations of length $n-1$. This map is used to give a new characterization, in terms of pattern avoidance, of almost-increasing cycles. Additionally, we use this bijection to enumerate several statistics on almost-increasing cycles. Such statistics include descents, inversions, peaks and excedances, as well as the newly defined statistic called low non-inversions. Furthermore, we refine the enumeration of unimodal permutations by descents, inversions and inverse valleys. We conclude this paper with a theorem that characterizes the standard cycle notation of almost-increasing permutations.
@article{10_37236_6954,
author = {Kassie Archer and L.-K. Lauderdale},
title = {Unimodal permutations and almost-increasing cycles},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6954},
zbl = {1372.05001},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6954/}
}
TY - JOUR
AU - Kassie Archer
AU - L.-K. Lauderdale
TI - Unimodal permutations and almost-increasing cycles
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/6954/
DO - 10.37236/6954
ID - 10_37236_6954
ER -
