Partial transpose of permutation matrices

Discrete Applied Mathematics, 2013

In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.

Applied Mathematics and Computation, 2015

The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the positive integer n is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of all n × n bipartite graphs.

Journal of Combinatorics, 2011

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are q-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a q-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

arXiv: Combinatorics, 2017

We compute the number of ways a given permutation can be written as a product of exactly $k$ transpositions. We express this number as a linear combination of explicit geometric sequences, with coefficients which can be computed in many particular cases. Along the way we prove several symmetry properties for matrices associated with bipartite graphs, as well as some general (likely known) properties of Young diagrams. The methods involve linear algebra, enumeration of border strip tableau, and a differential operator on symmetric polynomials.

Symmetry, 2023

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

arXiv: General Mathematics, 2018

This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation matrix and the cycle factorization of a permutation matrix or monomial matrix.

Zeitschrift für Operations Research, 1986

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors 6, recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

Ejc, 1995

A problem on maximal clones in universal algebra leads to the natural concept of orthogonal orders and their characterization. Two (partial) orders on the same set $P$ are orthogonal if they share only trivial endomorphisms i.e. if the identity selfmap of $P$ is the sole non-constant selfmap preserving (i.e. compatible with) both orders. We start with a neat and easy characterization of orthogonal pairs of chains (i.e. linear or total orders) and then proceed to the study of the number $q(k)$ of chains on $\{0,1, \ldots, k-1\}$ orthogonal to the natural chain $0<1\ldots<k-1$ . We obtain a recurrence formula for $q(k)$ and prove that the ratio $q(k)/k!$

Advances in Applied Mathematics, 2007

Consider S n , the symmetric group on n letters, and let maj π denote the major index of π ∈ S n. Given positive integers k, l and nonnegative integers i, j, define m k,l n (i, j) := #{π ∈ S n : maj π ≡ i (mod k) and maj π −1 ≡ j (mod l)} We give a bijective proof of the following result which had been previously proven by algebraic methods: If k, l are relatively prime and at most n then m k,l n (i, j) = n! kl which, surprisingly, does not depend on i and j. Equivalently, if m k,l n (i, j) is interpreted as the (i, j)-entry of a matrix m k,l n , then this is a constant matrix under the stated conditions. This bijection is extended to show the more general result, that for d ≥ 1 and k, l relatively prime, the matrix m kd,ld n admits a block decompostion where each block is the matrix m d,d n /(kl). We also give an explicit formula for m n,n n , and show that if p is prime then m p,p np has a simple block decomposition. To prove these results, we use the representation theory of the symmetric group and certain restricted shuffles.

2014

There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations $w$ that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of $w$, and fillings of a diagram of $w$ are equinumerous for all permutations $w$. Second, we give a $q$-analogue of a result of Hultman-Linusson-Shareshian-Sj\"ostrand by showing that under a certain pattern condition the Poincar\'e polynomial for the Bruhat interval of $w$ essentially counts invertible matrices avoiding a diagram of $w$ over a finite field. In addition to our main results, we include at the end a number of open questions.