Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.3547

Partial transpose of permutation matrices

Qing-Hu Hou·Toufik Mansour·Simone Severini
View original source ↗Read source document ↗Export metadata ↓

Abstract

The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are equal to their partial transpose and the number of permutation matrices whose partial transpose is still a permutation. We solve these problems also when restricted to symmetric permutation matrices only.

Explore related subjects

math.COquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Qing-Hu Hou, Toufik Mansour, Simone Severini. 2008-03-22. Partial transpose of permutation matrices. https://arxiv.org/abs/0709.3547

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ore plus Turán

Ore in 1961 determined the maximum number of edges in graphs not containing a Hamiltonian cycle, and Turán in 1941 found the maximum number of edges in graphs not containing a $K_{r+1}$. Motivated by the work of Adamus in 2009 and Ferrero and Lesniak in 2018 on the maximum number of edges in $r$-partite non-Hamiltonian graphs, we find the maximum number of edges in $K_{r+1}$-free non-Hamiltonian graphs. We determine the extremal graphs, which are different for $n \ge 26$ with $r \ge 3$, for small values of $n$ with $r \in \{4,5\}$, and for all values of $n$ with $r=2$. In the first case we show that the same graphs achieve the maximum number of $t$-cliques for every $t \ge 2$. Both the edge and clique results are proved via extremal theorems for graphs satisfying a condition on low-degree vertices related to Pósa's theorem, which we apply more broadly in a second paper. Both extend from Hamiltonicity to chorded pancyclicity.

math.CO↗
arXiv2026

A Proof of Rubey's Lattice Conjecture

In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation $w$ and conjectured that the induced poset on reduced pipe dreams is a lattice. In this paper, we prove this conjecture. Our key tool is a new type of move operation $\mathcal{M}_{ij}$, defined as a composite of certain generalized ladder moves and then simplified in terms of swaps on a partition shape. We show that joins and meets exist in Rubey's poset by proving simple recursive formulas in terms of $\mathcal{M}_{ij}$ operations, so we refer to the poset as the Rubey lattice henceforth. In addition, we give an explicit criterion to determine if two elements of the Rubey lattice are comparable using an injective map from reduced pipe dreams to tableaux on the diagram of a permutation. The pipe dream tableaux construction also gives an exact formula for the maximum length of any chain in the Rubey lattice and bounds on the number of reduced pipe dreams for $w$, or equivalently the sum of the coefficients in the corresponding Schubert polynomial. Several open problems for further development of the Rubey lattice related to Markov processes, Schubert polynomials, and Grothendieck polynomials are given.

math.CO↗
arXiv2026

Hamiltonicity of inhomogeneous random graphs

We provide a complete characterization of those graphons $W$ for which the inhomogeneous random graph \(\G(n,W)\) is asymptotically almost surely Hamiltonian. The characterization involves three conditions. Two of them constitute the characterization of $\G(n,W)$ being a.a.s.\ connected, as was shown recently by Hladký and Viswanathan. The third condition captures a geometric obstacle which prevents $\G(n,W)$ from having perfect fractional matchings. A prominent feature of the positive direction of our proof is the use of weak* limits to translate the absence of the above geometric obstacle into favourable matching properties of a regularization of $\G(n,W)$.

math.CO↗