This article has been retracted: please see Elsevier policy on Article Correction, Retraction and Removal (https://www.elsevier.com/about/policies-and-standards/article-withdrawal).This article has been retracted at the request of the Editors. The article purports to prove that a finite lattice is distributive if and only if each element has a unique irredundant decomposition as a join of join-irreducible elements. While the necessity of the condition is well known, its sufficiency asserted in the paper is false. The 7-element lower-semimodular non-modular lattice provides a long-known counterexample. The Editors thank Jakob Süskind for bringing this error to their attention.
This paper is concerned with the hypercube coloring problem. Let χℓ(m) be the minimum number of colors needed to color the vertices of a hypercube with 2m vertices such that every two vertices with Hamming distance ≤ℓ have different colors. Using linear codes over Z2k (the ring of integers modulo 2k) and the generalized Gray map, we establish a connection between the parameters of linear codes over Z2k and χℓ(m). Furthermore, we obtain some new upper bounds on χℓ(m). In some cases, the new upper bounds are tighter than the previously best known results. In particular, we determine the exact values χ5(14)=28 and χ4(13)=27.
We study the distribution of mesh patterns of length 2 on layered permutations, extending recent work on the distribution of mesh patterns on arbitrary permutations. We classify all mesh patterns of length 2 into exactly 35 equidistribution classes and derive a formula for the distribution of each class. With one exception, the corresponding avoidance sequences appear in the Online Encyclopedia of Integer Sequences, revealing connections with the Fibonacci and Tribonacci numbers, certain integer compositions, and restricted words. We also refine several of our enumerative results by jointly tracking five classical permutation statistics: descents, left-to-right maxima and minima, and right-to-left maxima and minima.
The problem of sampling edge-colorings of graphs with maximum degree Δ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to Δ. Vizing's theorem guarantees the existence of a (Δ+1)-edge-coloring, raising the natural question of how to efficiently sample such edge-colorings. In this paper, we take an initial step toward addressing this question. Building on the approach of Dotan, Linial, and Peled (2020), we analyze a randomized algorithm for generating random proper (Δ+1)-edge-colorings, which in particular provides an algorithmic interpretation of Vizing's theorem. The idea is to start from an arbitrary non-proper edge-coloring with the desired number of colors and at each step, recolor one edge uniformly at random provided it does not increase the number of conflicting edges (a potential function will count the number of pairs of adjacent edges of the same color). We show that the algorithm almost surely produces a proper (Δ+1)-edge-coloring and propose several conjectures regarding its efficiency and the uniformity of the sampled colorings.
As an important class of finite geometric structures, intriguing sets have attracted extensive research attention. In this paper, we study tight sets, a type of intriguing set, in the polar space Q−(7,q). The embedding of Q(6,q) in Q−(7,q) yields a (q3+1)-tight set. Since explicit constructions of nontrivial i-tight sets in Q−(7,q) are relatively rare, we investigate the three orbits O1, O2, and O3 of PSL3(q) on Q−(7,q). We prove that, when q is a prime power satisfying q≡2(mod3), these orbits are tight sets with parameters q+1, q(q2−1), and q3(q−1), respectively. We further prove that O1, O2, and O1∪O2 are nontrivial, thereby obtaining new infinite families of nontrivial tight sets. Additionally, we use the constructed (q+1)-tight set in Q−(7,q) to construct a minimal linear code over Fq when q≥5.
