Mathematics

We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.

Using a permutation code introduced by Rakotondrajao and the second author, we associate with every set partition of [n] a permutation over [n] , thus defining a class of permutation whose size is the n -th Bell number. We characterize the permutations belonging to this class and we study the distribution of weak exceedances over these permutations, which turns out to be enumerated by the Stirling numbers of the second kind. We provide a direct bijection between our class of permutations and another equisized class of permutations introduced by Poneti and Vajnovszki.

We determine the anti-Ramsey numbers for paths. This confirms a conjecture posed by Erd?s, Simonovits and Sós in 1970s.

The anti-adjacency matrix of graphs is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping each row and each column only the distances equal to 1. The anti-adjacency eigenvalues of a graph are those of its anti-adjacency matrix. Employing a novel technique introduced by Haemers [Spectral characterization of mixed extensions of small graphs, Discrete Math. 342 (2019) 2760--2764], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalues. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to -2 and 0. We finally propose some problems for further study.

A palindromic composition of n is a composition of n which can be read the same way forwards and backwards. In this paper we define an anti-palindromic composition of n to be a composition of n which has no mirror symmetry amongst its parts. We then give a surprising connection between the number of anti-palindromic compositions of n and the so-called tribonacci sequence, a generalization of the Fibonacci sequence. We conclude by defining a new q-analogue of the Fibonacci sequence, which is related to certain equivalence classes of anti-palindromic compositions

Let w ??=( w 1 ,?? w n )??R n . We show that for any n ?? ?�ϵ≤1 , if #{ ξ ???�{0,1 } n :??ξ ??, w ?????}??2 ?�ϵn ??2 n for some ??�R , then #{??ξ ??, w ???? ξ ???�{0,1 } n }??2 O( ϵ ??n) . This exponentially improves the ϵ dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and W?grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.

This short paper presents characterisations of normal arc-transitive circulants and arc-transitive normal circulants, that is, for a connected arc-transitive circulant $\Gamma=\Cay(C,S)$, it is shown that 1. Aut(C,S) is transitive on S if and only if each element of S has order n; 2. Aut??�C if and only if S does not contain a coset of any subgroup. This completes the classification of arc-transitive circulants given by Li-Xia-Zhou.

A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.

Given an integer g?? and a weight vector w??Q n ??0,1 ] n satisfying 2g??+??w i >0 , let ? g,w denote the moduli space of n -marked, w -stable tropical curves of genus g and volume one. We calculate the automorphism group Aut( ? g,w ) for g?? and arbitrary w , and we calculate the group Aut( ? 0,w ) when w is heavy/light. In both of these cases, we show that Aut( ? g,w )?�Aut( K w ) , where K w is the abstract simplicial complex on {1,??n} whose faces are subsets with w -weight at most 1 . We show that these groups are precisely the finite direct products of symmetric groups. The space ? g,w may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space M ¯ ¯ ¯ ¯ ¯ ¯ g,w . Following the work of Massarenti and Mella on the biregular automorphism group Aut( M ¯ ¯ ¯ ¯ ¯ ¯ g,w ) , we show that Aut( ? g,w ) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

In this paper, we reformulate the Bakry-?mery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.