We give a new semi-combinatorial proof for the equality of the number of ballot permutations of length $n$ and the number of odd order permutations of length $n$, which was originally proven by Bernardi, Duplantier and Nadeau. Spiro conjectures that the descent number of ballot permutations and certain cyclic weights of odd order permutations of the same length are equi-distributed. We present a bijection to establish a Toeplitz property for ballot permutations with any fixed number of descents, and a Toeplitz property for odd order permutations with any fixed cyclic weight. This allows us to refine Spiro's conjecture by tracking the neighbors of the largest letter in permutations.
@article{10_37236_9298,
author = {David G.L. Wang and Jerry J.R. Zhang},
title = {A {Toeplitz} property of ballot permutations and odd order permutations},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {2},
doi = {10.37236/9298},
zbl = {1454.05010},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9298/}
}
TY - JOUR
AU - David G.L. Wang
AU - Jerry J.R. Zhang
TI - A Toeplitz property of ballot permutations and odd order permutations
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/9298/
DO - 10.37236/9298
ID - 10_37236_9298
ER -
%0 Journal Article
%A David G.L. Wang
%A Jerry J.R. Zhang
%T A Toeplitz property of ballot permutations and odd order permutations
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/9298/
%R 10.37236/9298
%F 10_37236_9298
David G.L. Wang; Jerry J.R. Zhang. A Toeplitz property of ballot permutations and odd order permutations. The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/9298
