arXiv subjects

Ricky X. F. Chen

Publications and source records attributed to Ricky X. F. Chen.

26 records · Page 2Linked to original sources

arXiv2017

On a lower bound for sorting signed permutations by reversals

Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,\ldots -n,n,n-1,\ldots 1)$ and let $$ \tilde{s}=(0,a_1,a_2,\ldots a_n,-a_n,-a_{n-1},\ldots -a_1) $$ be any long cycle on the set $\{-n,-n+1,\ldots 0,1,\ldots n\}$. Then, $n$ and $a_n$ are always in the same cycle of the product $p\tilde{s}$. Furthermore, we show the new lower bound via plane permutations can be interpreted as the topological genera of orientable surfaces associated to signed permutations.

math.CO↗

arXiv2017

New formulas counting one-face maps and Chapuy's recursion

In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied by polynomials of the form $p(E)f(x)$, we easily establish the recursion obtained by Chapuy for $A(n,g)$. Moreover, we give a simple combinatorial interpretation for the Harer-Zagier recurrence.

math.CO↗

arXiv2016

Plane permutations and applications to a result of Zagier-Stanley and distances of permutations

In this paper, we introduce plane permutations, i.e. pairs $\mathfrak{p}=(s,π)$ where $s$ is an $n$-cycle and $π$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $π$) and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plane permutations obtained by permuting their diagonal-blocks. We establish basic properties of plane permutations and study transpositions and exceedances and derive various enumerative results. In particular, we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing Chapuy's recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of $n$-cycles $ω$, for which the product $ω(1~2~\cdots ~n)$ has exactly $k$ cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.

math.CO↗

arXiv2016

On the local genus distribution of graph embeddings

The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each vertex of $G$. In this paper, we study the following problem: given a genus $g$ embedding $ε$ of the graph $G$ and a vertex of $G$, how many different ways of reembedding the vertex such that the resulting embedding $ε'$ is of genus $g+Δg$? We give formulas to compute this quantity and the local minimal genus achieved by reembedding. In the process we obtain miscellaneous results. In particular, if there exists a one-face embedding of $G$, then the probability of a random embedding of $G$ to be one-face is at least $\prod_{ν\in V(G)}\frac{2}{deg(ν)+2}$, where $deg(ν)$ denotes the vertex degree of $ν$. Furthermore we obtain an easy-to-check necessary condition for a given embedding of $G$ to be an embedding of minimum genus.

math.CO↗

arXiv2015

Narayana polynomials and some generalizations

In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides combinatorial proofs for a bijective problem in Stanley's collection "Bijective Proof Problems", a new formula for the Narayana polynomials as well as a new expression for the Harer-Zagier formula enumerating unicellular maps, in a unified way. Furthermore, we identify a class of plane trees, whose enumeration is closely connected to the Schröder numbers. Many other binomial identities are presented as well.

math.CO↗

arXiv2015

On plane permutations

In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $ω$, for which $ω(12\cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It turns out that plane permutations allow to study transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Novel connections between these different distance problems are established via plane permutations.

math.CO↗

arXiv2015

A simple framework on sorting permutations

In this paper we present a simple framework to study various distance problems of permutations, including the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. These problems are very important in the study of the evolution of genomes. We give a general formulation for lower bounds of the transposition and block-interchange distance from which the existing lower bounds obtained by Bafna and Pevzner, and Christie can be easily derived. As to the reversal distance of signed permutations, we translate it into a block-interchange distance problem of permutations so that we obtain a new lower bound. Furthermore, studying distance problems via our framework motivates several interesting combinatorial problems related to product of permutations, some of which are studied in this paper as well.

math.CO↗

arXiv2015

Variation of the local topological structure of graph embeddings

The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each vertex of $G$. In this paper, we study the following problem: given a genus $g$ embedding $\mathbb{E}$ of the graph $G$, if we randomly rearrange the edges around a vertex, i.e., re-embedding, what is the probability of the resulting embedding $\mathbb{E}'$ having genus $g+Δg$? We give a formula to compute this probability. Meanwhile, some other known and unknown results are also obtained. For example, we show that the probability of preserving the genus is at least $\frac{2}{deg(v)+2}$ for re-embedding any vertex $v$ of degree $deg(v)$ in a one-face embedding; and we obtain a necessary condition for a given embedding of $G$ to be an embedding with the minimum genus.

math.CO↗