相关论文: Pattern frequency sequences and internal zeros

We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern…

In this paper, some zeros and non-zeros in the character tables of symmetric groups are displayed in the partition forms. In particular, more zeros of self conjugate partitions beside odd permutations are heavily investigated.

In this paper we propose numerical measures for evaluating the aesthetic interest of simple patterns. The patterns consist of elements (symbols, pixels, etc.) in regular square arrays. The measures depend on two characteristics of the…

Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed…

Let m(a,b) and M(a,b,c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a,c),m(a,b),m(b,c)) = M(a,b,c) for all a, b, c > 0. If m is strict and isotone, then we show that there exists a unique M which is type 1…

We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…

Let $\{U_n\}_{n \geqslant 0}$ and $\{G_m\}_{m \geqslant 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as…

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio…

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…

We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence,…

We present some results on the proportion of permutations of length $n$ containing certain mesh patterns as $n$ grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and…

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

We obtain an explicit formula for the number of permutations of [n] that avoid the barred pattern bar{1}43bar{5}2. A curious feature of its counting sequence, 1, 1, 2, 5, 14, 43, 145, 538, 2194,..., is that the displayed terms agree with…

A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…

Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1…

We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences…

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…