TY - JOUR
ID - 22243
TI - Combinatorial parameters on bargraphs of permutations
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Mansour, Toufik
AU - Shattuck, Mark
AD - Department of Mathematics, University of Tennessee, Knoxville, TN, USA
AD - Mathematics Department, University of Tennessee, Knoxville, TN, USA
Y1 - 2018
PY - 2018
VL - 7
IS - 2
SP - 1
EP - 16
KW - combinatorial statistic
KW - $q$-generalization
KW - bargraph
KW - permutations
DO - 10.22108/toc.2017.102359.1483
N2 - In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $\mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.
UR - https://toc.ui.ac.ir/article_22243.html
L1 - https://toc.ui.ac.ir/article_22243_ee9a92039072d73f603a278c71ef4387.pdf
ER -