Related papers: Adjacent q-cycles in permutations
It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…
We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…
The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…
We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Mikl\'{o}s…
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…
From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.
The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend…
We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a…
We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We obtain a generating function for the case of $q=312$ (equivalently, $q=231$), we prove that if $q$ is monotone increasing, then above a certain length,…
It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with…
It is well-known that any permutation can be written as a product of two involutions. We provide an explicit formula for the number of ways to do so, depending only on the cycle type of the permutation. In many cases, these numbers are sums…
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…
We study the total number of occurrences of several vincular (also called generalized) patterns and other statistics, such as the major index and the Denert statistic, on permutations avoiding a pattern of length 3, extending results of…
We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued…
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in…
The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities:…
We give analogues in the finite general linear group of two elementary results concerning long cycles and transpositions in the symmetric group: first, that the long cycles are precisely the elements whose minimum-length factorizations into…
We define integer multimodal sequences, which are generalizations of unimodal sequences having multiple local peaks of equal size. The generating functions for multimodal sequences represent novel types of $q$-series that combine generating…
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which…