arXiv subjects

Alice L. L. Gao

Publications and source records attributed to Alice L. L. Gao.

22 records · Page 2Linked to original sources

arXiv2016

On 132-representable Graphs

A graph $G = (V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Word-representable graphs are the subject of a long research line in the literature initiated in \cite{KP}, and they are the main focus in the recently published book \cite{KL}. A word $w=w_1\cdots w_{n}$ avoids the pattern $132$ if there are no $1\leq i_1<i_2<i_3\leq n$ such that $w_{i_1}<w_{i_3}<w_{i_2}$. The theory of patterns in words and permutations is a fast growing area discussed in \cite{HM,Kit}. A research direction suggested in \cite{KL} is in merging the theories of word-representable graphs and patterns in words. Namely, given a class of pattern-avoiding words, can we describe the class of graphs represented by the words? Our paper provides the first non-trivial results in this direction. We say that a graph is 132-representable if it can be represented by a 132-avoiding word. We show that each 132-representable graph is necessarily a circle graph. Also, we show that any tree and any cycle graph are 132-representable, which is a rather surprising fact taking into account that most of these graphs are non-representable in the sense specified, as a generalization of the notion of a word-representable graph, in \cite{JKPR}. Finally, we provide explicit 132-avoiding representations for all graphs on at most five vertices, and also describe all such representations, and enumerate them, for complete graphs.

math.CO↗

arXiv2016

On pattern avoiding indecomposable permutations

Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating $12\cdots k$-avoiding indecomposable permutations for $k\geq 3$. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies.

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arXiv2016

Avoiding vincular patterns on alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1 w_3 \cdots$ (when the word is up-down) or $w_1>w_2 w_4<\cdots$ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers. However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in question into equivalence classes. We provide a combinatorial argument to show that the number of equivalence classes is given by the Catalan numbers, which induces an alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}. Further, we extend the enumerative results in~\cite{GKZ} to the case of alternating words avoiding a vincular pattern of length 3. We show that it is sufficient to enumerate up-down words of even length avoiding the consecutive pattern $\underline{132}$ and up-down words of odd length avoiding the consecutive pattern $\underline{312}$ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.

math.CO↗

arXiv2016

Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue

In this paper, we confirm a conjecture of Laborde-Zubieta on the enumeration of corners in tree-like tableaux. Our proof is based on Aval, Boussicault and Nadeau's bijection between tree-like tableaux and permutation tableaux, and Corteel and Nadeau's bijection between permutation tableaux and permutations. This last bijection sends a corner in permutation tableaux to an ascent followed by a descent in permutations, this enables us to enumerate the number of corners in permutation tableaux, and thus to completely solve L.-Z.'s conjecture. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce an ($a$,$b$)-analogue of this enumeration, and explain the implications on the PASEP.

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