Advances in Applied Mathematics最新文献

Let $\alpha =a_1{a}_2\dots a_n$ be a sequence of nonnegative integers. The ascent set of α, Asc α, consists of all indices k where $a_{k+1}>a_k$. An ascent sequence is α where the growth of the $a_k$ is bounded by the elements of Asc α. These sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev and have many wonderful properties. In particular, they are in bijection with unlabeled $(2+2)$-free posets, permutations avoiding a particular bivincular pattern, certain upper-triangular nonnegative integer matrices, and a class of matchings. A weak ascent of α is an index k with $a_{k+1}\ge a_k$ and weak ascent sequences are defined analogously to ascent sequences. These were studied by Bényi, Claesson and Dukes and shown to have analogous equinumerous sets. Given a nonnegative integer d, we define a difference d ascent to be an index k such that $a_{k+1}>a_k-d$. We study the properties of the corresponding d-ascent sequences, showing that some of the maps from the weak case can be extended to bijections for general d while the extensions of others continue to be injective (but not surjective). We also make connections with other combinatorial objects such as rooted duplication trees and restricted growth functions.

Kirchhoff index is an electrical network-theoretic invariant which is defined as the sum of effective resistances between all pairs of vertices. As a robustness measure of simplicial networks, a simplicial analogue of the Kirchhoff index is defined to be the sum of simplicial effective resistances for all subsets of vertices of size dimension plus one. In this paper, we investigate the Kirchhoff index of random simplicial complexes as a generalization of random graphs. We present a formula for the expectation of the random variable and show how it concentrates around the expectation. We also perform numerical experiments revealing that the expectation and the fluctuation are still valid for realizations of the random simplicial Kirchhoff index.

Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Here, we focus on the stairs2 balance index for rooted binary trees, which was first introduced in the context of viral phylogenetics but has not been fully analyzed from a mathematical viewpoint yet. While it is known that the caterpillar tree uniquely minimizes the stairs2 index for all leaf numbers and the fully balanced tree uniquely maximizes the stairs2 index for leaf numbers that are powers of two, understanding the maximum value and maximal trees for arbitrary leaf numbers has been an open problem in the literature. In this note, we fill this gap by showing that for all leaf numbers, there is a unique rooted binary tree maximizing the stairs2 index. Additionally, we obtain recursive and closed expressions for the maximum value of the stairs2 index of a rooted binary tree with n leaves.

Combinatorial enumeration of various RNA secondary structures and protein contact maps is of great interest for both combinatorists and computational biologists. Counting protein contact maps is much more difficult than that of RNA secondary structures due to the significant higher vertex degree. The state of art upper bound for vertex degree in previous works is two. This paper proposes a solution for counting general stacks with arbitrary vertex degree upper bound. By establishing a bijection between such general stacks and m-regular Λ-avoiding DLU-paths, and counting these pattern avoiding lattice paths, we obtain a unified system of equations for the generating functions of the number of general stacks. We further show that previous enumeration results for RNA secondary structures and linear stacks of protein contact maps can be derived from the equations for general stacks as special cases.

The authors regret that there was a mistake in [1, Definition 26] with our new basis axiom (nB3). We explain and correct this mistake here.

Let $\left(a\right(n):n\in N_0)$ denote an automatic sequence. Previous research on infinite products involving automatic sequences has mainly dealt with identities for products as in ${\prod }_{n}R{\left(n\right)}^{a\left(n\right)}$ for rational functions $R\left(n\right)$. This inspires the development of techniques for evaluating ${\prod }_{n}f{\left(n\right)}^{a\left(n\right)}$ more generally, for functions $f\left(n\right)$ that are not rational functions. This leads us to apply Euler's product expansion for the Γ-function together with recursive properties of $a\left(n\right)$ to obtain identities as in ${\prod }_{n}f{(n,z)}^{a\left(n\right)}=\Gamma (z+1)$, and this is motivated by how the equivalent series identity ${\sum }_{n}a\left(n\right)\ln f(n,z)=\ln \Gamma (z+1)$ could be applied in relation to the remarkable results due to Gosper on the integration of $\ln \Gamma (z+1)$. We succeed in applying this approach, using Gosper's integration identities, to obtain new infinite products that we evaluate in terms of the Glaisher–Kinkelin constant A and that involve the Thue–Morse sequence, the period-doubling sequence, and the regular paperfolding sequence. A byproduct of our method gives us a way of generalizing a Dirichlet series identity due to Allouche and Sondow, and we also explore applications related to a product evaluation due to Gosper involving A.

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X, and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through $X_k$.

In particular, we show how to take advantage of homogeneous sets of G (those $S\subseteq V\left(G\right)$ such that each vertex of $V\left(G\right)﹨S$ is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1\bigsqcup S_2\subseteq V\left(G\right)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.

Let $A=(a_1,a_2,\dots ,a_n)$ be relative prime positive integers with $a_i\ge 2$. The Frobenius number $g\left(A\right)$ is the greatest integer not belonging to the set $\left\{{\sum }_{i=1}^n{a}_i{x}_i\right|x_i\in N\}$. The general Frobenius problem includes the determination of $g\left(A\right)$ and the related Sylvester number $n\left(A\right)$ and Sylvester sum $s\left(A\right)$. We present a combinatorial approach to the Frobenius problem. Basically, we transform the problem into an easier optimization problem. If the new problem can be solved explicitly, then we obtain a formula for $g\left(A\right)$. We illustrate the idea by giving concise proofs and extensions of several existing formulas, as well as new formulas for $g\left(A\right),n\left(A\right),s\left(A\right)$. Moreover, we give a generating function approach to $n\left(A\right),s\left(A\right)$, and even to the more general Sylvester power sum.

A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if G is a graph on n vertices with at least $\lfloor n^2/4\rfloor +1$ edges, then G contains at least $\lfloor n/2\rfloor$ triangles. We prove a spectral version of Erdős–Rademacher's theorem. Moreover, Mubayi (2010) [28] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs.

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their $Q$-linear independence.