Advances in Applied Mathematics最新文献

We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of $(\mathrm{exc},\mathrm{fix})$ over $S_n$ and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With the grammar in hand, we demonstrate the advantage of the grammatical calculus in deriving the generating functions, where the constant property plays a substantial role. On the grounds of left successions of a permutation, we present a grammatical treatment of the joint distribution investigated by Roselle. Moreover, we obtain a left succession analogue of the Diaconis-Evans-Graham theorem, exemplifying the idea of a grammar assisted bijection. The grammatical labelings give rise to an equidistribution of $(\mathrm{jump},\mathrm{des})$ and $(\mathrm{exc},\mathrm{drop})$ restricted to the set of left successions and the set of fixed points, where jump is defined to be the number of ascents minus the number of left successions.

The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set ${\check{\Lambda }}^q$ of strict integer partitions (i.e., with unequal parts) into perfect q-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $〈N〉$ and $〈M〉$ controlling the expected weight and length, respectively. We study “short” partitions, where the parameter $〈M〉$ is either fixed or grows slower than for typical partitions in ${\check{\Lambda }}^q$. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $〈M〉$ and a limit shape result in the case of slow growth of $〈M〉$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyze their performance.

In the present paper, we introduce a new approach and use it to prove that the maximum number of triangles in a $C_5$-free graph on n vertices is at most $\frac{1}{2\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, improving an estimate of Ergemlidze and Methuku [4]. We also show that the maximum size of an induced-$C_4$-free and $C_5$-free graph on n vertices is at most $\frac{1}{\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, also improving an estimate of Ergemlidze and Methuku [4].

We determine the maximal non-central hyperplane sections of the $l_1^n$-ball if the fixed distance of the hyperplane to the origin is between $\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{2}}$. This adds to a result of Liu and Tkocz who considered the distance range between $\frac{1}{\sqrt{2}}$ and 1. For $n\ge 4$, the maximal sections are parallel to the $(n-1)$-dimensional coordinate planes. We also study non-central sections of the complex $l_{\infty }^2$-ball, where the formulas are more complicated than in the real case. Also, the extrema are partially different compared to the real case.

We introduce a kind of $(p,q,t)$-Catalan numbers of Type A by generalizing the J-type continued fraction formula, we prove that the corresponding expansions could be expressed by the polynomials counting permutations on $S_n\left(321\right)$ by various descent statistics. Moreover, we introduce a kind of $(p,q,t)$-Catalan numbers of Type B by generalizing the J-type continued fraction formula, we prove that the Taylor coefficients and their γ-coefficients could be expressed by the polynomials counting permutations on $S_n(3124,4123,3142,4132)$ by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.

The boolean elements of a Coxeter group have been characterized and shown to possess many interesting properties and applications. Here we introduce “prism permutations,” a generalization of those elements, characterizing the prism permutations equivalently in terms of their reduced words and in terms of pattern containment. As part of this work, we introduce the notion of “calibration” to permutation patterns.

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.