Combinatorial parameters on bargraphs of permutations
Toufik
Mansour
Department of Mathematics, University of Tennessee, Knoxville, TN, USA
author
Mark
Shattuck
Mathematics Department, University of Tennessee, Knoxville, TN, USA
author
text
article
2018
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
2
no.
2018
1
16
http://toc.ui.ac.ir/article_22243_ee9a92039072d73f603a278c71ef4387.pdf
dx.doi.org/10.22108/toc.2017.102359.1483
The log-convexity of the fubini numbers
Qing
Zou
The University of Iowa
author
text
article
2018
eng
Let $f_n$ denotes the $n$th Fubini number. In this paper, first we give upper and lower bounds for the Fubini numbers $f_n$. Then the log-convexity of the Fubini numbers has been obtained. Furthermore we also give the monotonicity of the sequence $\{\sqrt[n]{f_n}\}_{n\ge 1}$ by using the aforementioned bounds.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
2
no.
2018
17
23
http://toc.ui.ac.ir/article_21835_8b52d6cf1daabf7e0e9be379112846e3.pdf
dx.doi.org/10.22108/toc.2017.104212.1496
Solution to the minimum harmonic index of graphs with given minimum degree
Meili
Liang
Guangdong University of Foreign Studies
author
Bo
Cheng
Guangdong University of Foreign Studies
author
Jianxi
Liu
Guangdong University of Foreign Studies
author
text
article
2018
eng
The harmonic index of a graph $G$ is defined as $ H(G)=\sum\limits_{uv\in E(G)}\frac{2}{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. Let $\mathcal{G}(n,k)$ be the set of simple $n$-vertex graphs with minimum degree at least $k$. In this work we consider the problem of determining the minimum value of the harmonic index and the corresponding extremal graphs among $\mathcal{G}(n,k)$. We solve the problem for each integer $k (1\le k\le n/2)$ and show the corresponding extremal graph is the complete split graph $K_{k,n-k}^*$. This result together with our previous result which solve the problem for each integer $k (n/2 \le k\le n-1)$ give a complete solution of the problem.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
2
no.
2018
25
33
http://toc.ui.ac.ir/article_22272_28d4f6f37d2867d952c1398e234888f8.pdf
dx.doi.org/10.22108/toc.2017.101076.1462
On matrix and lattice ideals of digraphs
Hamid
Damadi
Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.
author
Farhad
Rahmati
Amirkabir University of Technology
author
text
article
2018
eng
Let $\textit{G}$ be a simple, oriented connected graph with $n$ vertices and $m$ edges. Let $I(\textbf{B})$ be the binomial ideal associated to the incidence matrix \textbf{B} of the graph $G$. Assume that $I_L$ is the lattice ideal associated to the rows of the matrix $\textbf{B}$. Also let $\textbf{B}_i$ be a submatrix of $\textbf{B}$ after removing the $i$-th row. We introduce a graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(\textbf{B})=I(\textbf{B}_i)$ and $I(\textbf{B}_i)=I_L$. After that we introduce another graph theoretical criterion for $G$ which is a sufficient and necessary condition for $I(\textbf{B})=I_L$. It is shown that the heights of $I(\textbf{B})$ and $I(\textbf{B}_i)$ are equal to $n-1$ and the dimensions of $I(\textbf{B})$ and $I(\textbf{B}_i)$ are equal to $m-n+1$; then $I(\textbf{B}_i)$ is a complete intersection ideal.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
2
no.
2018
35
46
http://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdf
dx.doi.org/10.22108/toc.2017.105701.1510
Reduced zero-divisor graphs of posets
Deiborlang
Nongsiang
North Eastern Hill University
author
Promode
Saikia
North Eastern Hill University
author
text
article
2018
eng
This paper investigates properties of the reduced zero-divisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a non-annihilator prime ideal. We also classify all posets whose reduced zero-divisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.
Transactions on Combinatorics
University of Isfahan
2251-8657
7
v.
2
no.
2018
47
54
http://toc.ui.ac.ir/article_22311_893dce7cc938e8e23dd5defcadb2c102.pdf
dx.doi.org/10.22108/toc.2018.55164.1417