Simultaneous avoidance of large squares and fractional powers in infinite binary words

Simultaneous avoidance of generalized patterns

In [BabStein] Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case…

同时避免长度为4的模式在上升序列中的出现

Simultaneous avoidance of length-4 patterns in ascent sequences

研究了上升序列中避免长度为4的五个模式（0101, 0102, 0112, 0120, 0121）的计数问题，发现这些模式避免类分为16个Wilf等价类，并揭示了与Catalan数、Fibonacci数等经典序列以及多项式公式和有理生成函数的联系。

New results on pseudosquare avoidance

We start by considering binary words containing the minimum possible numbers of squares and antisquares (where an antisquare is a word of the form $x \overline{x}$), and we completely classify which possibilities can occur. We consider avoiding $x p(x)$, where $p$ is any permutation of the underlying alphabet, and $x t(x)$, where $t$ is any transformation of the underlying alphabet. Finally, we prove the existence of an infinite binary word simultaneously avoiding all occurrences of $x h(x)$ for every nonerasing morphism $h$ and all sufficiently large words $x$.

卷曲数变换中的重复避免

Repetition Avoidance in Curling-Number Transforms

该研究针对字及其卷曲数变换的重复避免问题，通过Thue-Morse形态构造与穷举搜索得出：4是允许同时实现无限无重叠性的最小字母表大小，相关结果经Walnut验证。

Non-Contiguous Pattern Avoidance in Binary Trees

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding t…

The Simultaneous Fractional Dimension of Graph Families

A subset $S$ of the vertices $V$ of a connected graph $G$ resolves $G$ if no two vertices of $V$ share the same list of distances (shortest-path metric) with respect to the vertices of $S$ listed in a given order. The choice of such an $S$ in $V$ amounts to selecting a binary valued function $g$, said to be a resolving function, on $V$. The notion of a fractional resolving function is obtained by relaxing the codomain of $g$ to be the unit interval. Let $|g|=\sum_{v\in V}g(v)$. Given a finite collection $\mathcal{G}$ of connected graphs on a common vertex set $V$, the simultaneous metric dimen…