We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence ad0M,adM,ad2M,… of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.

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×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.

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

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.

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].

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 no(1). 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.

Rowmotion is a certain well-studied bijective operator on the distributive lattice J(P) of order ideals of a finite poset P. We introduce the rowmotion Markov chainMJ(P) by assigning a probability px to each x∈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 pj to each join-irreducible element j of L and use these probabilities to construct a rowmotion Markov chain ML. Under the assumption that each probability pj is strictly between 0 and 1, we prove that ML 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 ML 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 Tλ(τ) and STλ(τ) 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 λ/μ for any skew diagram λ/μ. The equidistribution enables us to show that the peak set is equidistributed over Tλ(12⋯kτ) (resp. STλ(12⋯kτ)) and Tλ(k⋯21τ) (resp. STλ(k⋯21τ)) for any Young diagram λ and any permutation τ of {k+1,k+2,…,k+m} with k,m≥1. Our results are refinements of the result of Backelin-West-Xin which states that |Tλ(12⋯kτ)|=|Tλ(k⋯21τ)| and the result of Bousquet-Mélou and Steingrímsson which states that |STλ(12⋯kτ)|=|STλ(k⋯21τ)|. As applications, we are able to•confirm a recent conjecture posed by Yan-Wang-Zhou which asserts that the peak set is equidistributed over 12⋯kτ-avoiding involutions and k⋯21τ-avoiding involutions;•prove that alternating involutions avoiding the pattern 12⋯kτ are equinumerous with alternating involutions avoiding the pattern k⋯21τ, paralleling the result of Backelin-West-Xin for permutations, the result of Bousquet-Mélou and Steingrímsson for involutions, and the result of Yan for alternating permutations.

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.