Related papers: AZ-identities and Strict 2-part Sperner Properties…
Sperner theory is one of the most important branches in extremal set theory. It has many applications in the field of operation research, computer science, hypergraph theory and so on. The LYM property has become an important tool for…
The AZ identity is a generalization of the LYM-inequality. In this paper, we will give a generalization of the AZ identity.
By using nonstandard analysis, we prove embeddability properties of difference sets $A-B$ of sets of integers. (A set $A$ is "embeddable" into $B$ if every finite configuration of $A$ has shifted copies in $B$.) As corollaries of our main…
Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a \emph{pircon} be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti's zircons. We…
Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner…
A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\H{o}s characterised the largest families of subsets of…
An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's…
The width of a poset is the size of its largest antichain. Sperner's theorem states that $(2^{[n]},\subset)$ is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends…
Some inequalities in 2-inner product spaces generalizing Bessel's result that are similar to the Boas-Bellman inequality from inner product spaces, are given. Applications for determinantal integral inequalities are also provided.
The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid…
We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number…
We classify finite posets with a particular sorting property, generalizing a result for rectangular arrays. Each poset is covered by two sets of disjoint saturated chains such that, for any original labeling, after sorting the labels along…
We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and…
We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the hyperoctahedral…
In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\text{ZZ}(\boldsymbol{S},x)$ of a Kekul\'ean regular $m$-tier strip $\boldsymbol{S}$ of length $n$ and the extended strict order…
In Heintz-Schnorr (1982), the authors introduced the notion of correct test sequence and since then it has been widely used to design probabilistic algorithms for Polynomial Equality Test. The aim of this manuscript is to study the…
We consider any fixed $d\in\mathbb{Z}_{>0}$ number of second class particles in the asymmetric simple exclusion process (ASEP), constructed via a basic coupling of two ASEPs. We give the joint distribution of the positions of the second…
The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the…
In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function.…
We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are…