Andrey O. Matveev
We establish monotone bijections between the Farey sequences of order m and the halfsequences of Farey subsequences associated with the rank m elements of the Boolean lattice of subsets of a 2m-set. We also present a few related combinatorial identities.
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We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
We describe monotone maps between subsequences of the Farey sequences.
In this note, we study a family of subgraphs of the Farey graph, denoted as $\mathcal{F}_N$ for every $N\in\mathbb{N}.$ We show that $\mathcal{F}_N$ is connected if and only if $N$ is either equal to one or a prime power. We introduce a class of continued fractions referred to as $\mathcal{F}_N$-continued fractions for each $N>1.$ We establish a relation between $\mathcal{F}_N$-continued fractions and certain paths from infinity in the graph $\mathcal{F}_N.$ We discuss existence and uniqueness of $\mathcal{F}_N$-continued fraction expansions of real numbers.
Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.
Let $F$ be a quadratic APN function of $n$ variables. The associated Boolean function $γ_F$ in $2n$ variables ($γ_F(a,b)=1$ if $a\neq{\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions) has the form $γ_F(a,b) = Φ_F(a) \cdot b + φ_F(a) + 1$ for appropriate functions $Φ_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $φ_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We summarize the known results and prove new ones regarding properties of $Φ_F$ and $φ_F$. For instance, we prove that degree of $Φ_F$ is either $n$ or less or equal to $n-2$. Based on computation experiments, we formulate a conjecture that degree of an…
The characterization of level sets of finite Boolean lattices as antichain cutsets, due to Rival and Zaguia, is seen to hold in all discrete semimodular lattices.
The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games it was only recently shown that the dimension can also be exponential. Here we show that this is also the case for complete simple games with two types of voters and for the Boolean dimension of general complete simple games, which was posed as an open problem.
The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: $A\subset B,C\subset D$. A diamond-free family in the $n$-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements $B$ and $C$ may or may not be related. There is a diamond-free family in the $n$-dimensional Boolean lattice of size $(2-o(1)){n\choose\lfloor n/2\rfloor}$. In this paper, we prove that any diamond-free family in the $n$-dimensional Boolean lattice has size at most $(2.25+o(1)){n\choose\lfloor n/2\rfloor}$. …
The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are so-called aggregation functions. We first explicitly classify the Lovász extensions of pseudo-Boolean functions according to their arity gap. Then we consider the class of order-preserving functions between partially ordered sets, and establish a similar explicit classification for this function class.
Note on the number of antichains in generalizations of the Boolean lattice
本文通过简短自足的论证给出了布尔格推广 $[t]^n$ 中反链数量的上界，并在 $t=3$ 时证明了目前最佳的上界，部分回答了 Moshkovitz 和 Shapira 提出的问题。
Efficient Evaluations of Weighted Sums over the Boolean Lattice inspired by conjectures of Berti, Corsi, Maspero, and Ventura
本文针对水波研究中提出的布尔格加权和猜想，给出了高效生成项与求闭式解的方法，给出首个猜想的新证明，并为第二个猜想提供了有力证据，且第二版中该猜想已被他人借助文中递推关系证明。
In this note we give a version of Hao Huang's proof of the sensitivity conjecture, shedding some light on the origin of the magical matrix $A$ in that proof. For the history of the subject and the importance of this conjecture to the study of boolean functions, we refer to the original paper. Here we only state the main result: Consider the boolean cube $Q_n=\{0,1\}^n$ as a graph, whose edges connect pairs of vertices differing in one coordinate. Then any its induced subgraph on greater than $2^{n-1}$ (the half) vertices has degree of some vertex at least $\sqrt{n}$.
In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof of Björner and Wachs' result \cite{BW} that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra --- that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is t…
Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulan…
The purpose of this expository note is to give the proof of a theorem of Bourgain with some additional details and updated notation. The theorem first appeared as an appendix to the breakthrough paper by Friedgut, \emph{Sharp Thresholds of graph properties and the $k$-SAT Problem}. Throughout, we use notation and definitions akin to those in O'Donnell's book, \emph{Analysis of Boolean Functions}.
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of squares of influences in $f$ is close to zero then $f$ must be noise sensitive. We show a quantitative version of this result which does not depend on $n$, and prove that it is tight for certain parameters. Our results hold also for a general product measure $μ_p$ on the discrete cube, as long as $\log 1/p \ll \log n$. We note that in [BKS], a quantitative relatio…
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