Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1604.03418

Asymptotic Behavior of the Expectation Value of Permanent Products, a Sequel

Paul Federbush
View original source ↗Read source document ↗Export metadata ↓

Abstract

Continuing the computations of the previous paper,[1], we calculate another approximation to the expectation value of the product of two permanents in the ensemble of 0-1 n x n matrices with like row and column sums equal r uniformly weighted. Here we consider the Bernoulli random matrix ensemble where each entry independently has a probability p=r/n of being one, otherwise zero. We denote the expectations of the approximation ensemble of [1] by E, and the expectations of the present approximation ensemble, the Bernoulli random matrix ensemble, by E*. One has for these lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E(perm_m(A))) -lim_{n to infinity} (1/n) ln(E*(perm_m(A))) ) = 0 and lim_{n to infinity} (1/n) ln(E(perm_m(A)perm_m'(A))) = lim_{n to infinity} (1/n) ln(E(perm_m(A))) + lim_{n to infinity} (1/n) ln(E(perm_m'(A))) Here and in all such formulas the subscripts m,m' are assumed proportional to n. It seems likely to us that lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E*(perm_m(A)perm_m'(A))) - lim_{n to infinity} (1/n) ln(E*(perm_m(A))) + - lim_{n to infinity} (1/n) ln(E*(perm_m'(A))) ) = 0 We believe: "E gives us the `correct' expectations in these equations, and E* is only `correct' in the r to infinity limit."

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paul Federbush. 2016-04-12. Asymptotic Behavior of the Expectation Value of Permanent Products, a Sequel. https://arxiv.org/abs/1604.03418

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

KEEP EXPLORING

Related papers

arXiv2026

On Perfect Divisibility of Bull-Free Graphs Without Long Paths

A graph $G$ is {\em perfectly divisible} if, for every induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. Chudnovsky and Sivaraman [J. Graph Theory \textbf{90} (2019) 54-60] proved that every ($P_5$, bull)-free graph is perfectly divisible, while Chen and Xu [Discrete Appl. Math. \textbf{372} (2025) 298-307] proved the same for ($P_7,C_5$, bull)-free graphs. We extend these results by proving that every ($P_8,C_5$, bull)-free graph is perfectly divisible and that, letting $F$ denote the Grötzsch graph, a ($P_6$, bull)-free graph is perfectly divisible if and only if it is $F$-free.

math.CO↗
arXiv2026

Covering graphs by isometric trees

A connected subgraph of a graph is isometric if it preserves distances. Recently, graphs admitting a vertex or edge covering by a small number of isometric paths have been studied. In this paper, we consider the analogous problem for isometric trees, focusing on the treewidth of graphs admitting a vertex or edge covering by a small number of such trees. Baste, De Meyer, Giocanti, Objois, and Picavet showed that for coverings by two isometric trees, the treewidth is bounded. We show that already for three isometric trees, the treewidth can be linear in the number of vertices. On the positive side, we show that for graphs of bounded degree coverable by a small number of isometric trees, the treewidth is sublinear in the number of vertices.

math.CO↗
arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗