使用+、-、×和/运算符构造的非同构算术表达式的数量

Permutations destroying arithmetic progressions in finite cyclic groups

A permutation πof an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (π(a),π(b),π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mo…

关于正整数排列中单调等差数列结果的一个推广

A Generalisation of a Result on Monotone Arithmetic Progressions in Permutations of the Positive Integers

该论文推广了正整数排列中避免特定单调等差数列的构造，证明对任意k≥1，存在排列避免长度为4且公差不被2^k整除的等差数列。

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We construct a permutation of the integers avoiding arithmetic progressions of length $6$. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length $4$, sharpening the lower bound of $\frac{1}{3}$ from (LeSaulnier and Vijay, 2011). In addition, we generalize several…

On the number of non-isomorphic (simple) $k$-gonal biembeddings of complete multipartite graphs

This article aims to provide exponential lower bounds on the number of non-isomorphic $k$-gonal biembeddings of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazici in 2020, we obtain that, for infinitely many values of $k$ and…

算术非非常通用排列

Arithmetic non-very generic arrangements

本文研究了判别排列中非非常通用交集的算术特征，并修正了之前关于二阶交集的结果。

Finite presentability and isomorphism of Cayley graphs of monoids

Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.

Solution group representations as quantum symmetries of graphs

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex- and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the f…

Two-Step and Three-Step Nilpotent Lie Algebras Constructed from Schreier Graphs

We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra. As an application, if we start with pairs of non-isomorphic Schreier graphs coming from Gassmann-Sunada triples, we prove that the pair of associated two-step nilpotent Lie algebras are always isometric. In contrast, we use a well-known pair of Schreier graphs to show that the associated three-step nilpotent extensions need not be isometric.