Related papers: A Survey of Alternating Permutations

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random…

Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length…

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down…

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\'ery sums). The study of non-alternating sums leads to an investigation of different types…

We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating…

This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations,…

We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.

A ballot permutation is a permutation {\pi} such that in any prefix of {\pi} the descent number is not more than the ascent number. In this article, we obtained a formula in close form for the multivariate generating function of {A(n,d,j)},…

Arc permutations, which were originally introduced in the study of triangulations and characters, have recently been shown to have interesting combinatorial properties. The first part of this paper continues their study by providing signed…

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if…

The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…

We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential…

Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural…

We give a new proof of Johnsonbaugh's refined error estimates of an alternating series based on an idea of R. M. Young. We also give a new proof of the error estimate and convergence of the Euler transform.

We study a modification of Kendall's tau-test, replacing his permutations of n different numbers by sequences of length n, where repetition is allowed. In particular, binary sequences are included. Random sequences can be tested.

In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…