Advances in Applied Mathematics最新文献

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection φ between binary trees and plane trees answers an open problem posed by Bai and Chen. This involution can be generalized to weakly increasing trees, which admits to merge two recent equidistributions found by Bai–Chen and Chen–Fu, respectively. The other one is constructed to answer a bijective problem on di-sk trees asked by Fu–Lin–Wang and can be generalized naturally to rooted labeled trees. This second involution combined with φ leads to a new statistic on plane trees whose distribution gives the Catalan's triangle. Moreover, a quadruple equidistribution on plane trees involving this new statistic is proved via a recursive bijection.

This article concerns the Orlicz-Minkowski problem for $p$-capacity for $1<p<n$. We use the flow method to obtain a new existence result of solutions to this problem by an approximation argument for general measures.

Despite their noted potential in polynomial-system solving, there are few concrete examples of Newton-Okounkov bodies arising from applications. Accordingly, in this paper, we introduce a new application of Newton-Okounkov body theory to the study of chemical reaction networks and compute several examples. An important invariant of a chemical reaction network is its maximum number of positive steady states Here, we introduce a new upper bound on this number, namely the ‘Newton-Okounkov body bound’ of a chemical reaction network. Through explicit examples, we show that the Newton-Okounkov body bound of a network gives a good upper bound on its maximum number of positive steady states. We also compare this Newton-Okounkov body bound to a related upper bound, namely the mixed volume of a chemical reaction network, and find that it often achieves better bounds.

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.

As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let $T_{\lambda }\left(\tau \right)$ and ${\mathrm{ST}}_{\lambda }\left(\tau \right)$ denote the set of τ-avoiding transversals and τ-avoiding symmetric transversals of a Young diagram λ, respectively. In this paper, we are mainly concerned with the distribution of the peak set and the valley set on standard Young tableaux and pattern avoiding transversals. In particular, we prove that the peak set and the valley set are equidistributed on the standard Young tableaux of shape $\lambda /\mu$ for any skew diagram $\lambda /\mu$. The equidistribution enables us to show that the peak set is equidistributed over $T_{\lambda }(12\cdots k\tau )$ (resp. ${\mathrm{ST}}_{\lambda }(12\cdots k\tau )$) and $T_{\lambda }(k\cdots 21\tau )$ (resp. ${\mathrm{ST}}_{\lambda }(k\cdots 21\tau )$) for any Young diagram λ and any permutation τ of $\{k+1,k+2,\dots ,k+m\}$ with $k,m\ge 1$. Our results are refinements of the result of Backelin-West-Xin which states that $\left|T_{\lambda }\right(12\cdots k\tau \left)\right|=\left|T_{\lambda }\right(k\cdots 21\tau \left)\right|$ and the result of Bousquet-Mélou and Steingrímsson which states that $\left|{\mathrm{ST}}_{\lambda }\right(12\cdots k\tau \left)\right|=\left|{\mathrm{ST}}_{\lambda }\right(k\cdots 21\tau \left)\right|$. As applications, we are able to

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  confirm a recent conjecture posed by Yan-Wang-Zhou which asserts that the peak set is equidistributed over