Advances in Applied Mathematics最新文献

Let m be a positive integer larger than 1, w be a finite word over $\{0,1,\cdots ,m-1\}$ and $a_{m;w}\left(n\right)$ represent the number of occurrences of the word w in the m-expansion of the non-negative integer n (mod m). In this article, we present an efficient algorithm for generating all sequences ${\left(a_{m;w}\right(n\left)\right)}_{n\in N}$; then, assuming that m is a prime number, we prove that all these sequences are m-uniformly but not purely morphic, except for words w satisfying $\left|w\right|=1$ and $w\ne 0$; finally, under the same assumption of m as before, we prove that the power series ${\sum }_{i=0}^{\infty }{a}_{m;w}\left(n\right)t^n$ is algebraic of degree m over $F_m\left(t\right)$.

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the threshold probability at which such graphs attain maximal connectivity. We slightly improve the lower bounds on the threshold probabilities, thus disproving two conjectures of Alon, Defant and Kravitz. We also improve the upper bound on the threshold probability in the case of random bipartite graphs, and obtain a tight bound up to a factor of $n^{o\left(1\right)}$. Further, we introduce a generalization of the notion of friends-and-strangers graphs in which vertices of the starting graphs are allowed to have multiplicities and obtain generalizations of previous results of Wilson and of Defant and Kravitz in this new setting.

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15].

Rowmotion is a certain well-studied bijective operator on the distributive lattice $J\left(P\right)$ of order ideals of a finite poset P. We introduce the rowmotion Markov chain $M_{J\left(P\right)}$ by assigning a probability $p_x$ to each $x\in P$ and using these probabilities to insert randomness into the original definition of rowmotion. More generally, we introduce a very broad family of toggle Markov chains inspired by Striker's notion of generalized toggling. We characterize when toggle Markov chains are irreducible, and we show that each toggle Markov chain has a remarkably simple stationary distribution.

We also provide a second generalization of rowmotion Markov chains to the context of semidistrim lattices. Given a semidistrim lattice L, we assign a probability $p_j$ to each join-irreducible element j of L and use these probabilities to construct a rowmotion Markov chain $M_L$. Under the assumption that each probability $p_j$ is strictly between 0 and 1, we prove that $M_L$ is irreducible. We also compute the stationary distribution of the rowmotion Markov chain of a lattice obtained by adding a minimal element and a maximal element to a disjoint union of two chains.

We bound the mixing time of $M_L$ for an arbitrary semidistrim lattice L. In the special case when L is a Boolean lattice, we use spectral methods to obtain much stronger estimates on the mixing time, showing that rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hästö. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the interior. The purpose of the following is to study this analogy in greater detail. We supplement the investigation of copolarity, considering, for example, conjugate faces. Then we deal with the question suggested by Minkowski's theorem, asking which measures are surface area measures of pseudo-cones with given recession cone. We provide a sufficient condition for possibly infinite measures and a special class of pseudo-cones.

We study divide-and-conquer recurrences of the form$f\left(n\right)=\alpha f(\lfloor \frac{n}{2}\rfloor )+\beta f(\lceil \frac{n}{2}\rceil )+g\left(n\right)(n⩾2),$ with $g\left(n\right)$ and $f\left(1\right)$ given, where $\alpha ,\beta ⩾0$ with $\alpha +\beta >0$; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on $g\left(n\right)$ that the solution f always satisfies a simple identity$f\left(n\right)=n^{{\log }_2(\alpha +\beta )}P({\log }_{2}n)-Q\left(n\right),$ where P is a periodic function and $Q\left(n\right)$ is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the continuity of the periodic function P are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies.

Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ-positive polynomials from γ-positive polynomials. We then prove the alternating γ-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored $2\times n$ Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B.

In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{\left(m\right)}$. We also establish one-to-one correspondence between all positive integers in the set $\{1,\cdots ,m^{n}n!\}$ and the elements of $G(m,1,n)$ by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for $G(m,1,n)$ by defining the inversion statistic on $G(m,1,n)$. Finally, we prove that the flag-major index is equi-distributed with this inversion statistic on $G(m,1,n)$. Therefore, the flag-major index is a Mahonian statistic on $G(m,1,n)$ with respect to the length function L.