Related papers: A Generating Function for the Distribution of Runs…
We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and…
Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in…
A parking function is a sequence $(a_1,\dots, a_n)$ of positive integers such that if $b_1\leq\cdots\leq b_n$ is the increasing rearrangement of $a_1,\dots,a_n$, then $b_i\leq i$ for $1\leq i\leq n$. In this paper we obtain some new results…
We determine the distributions of lengths of runs in random sequences of elements from a totally ordered set (total order) or partially ordered set (partial order). In particular, we produce novel formulae for the expected value, variance,…
Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…
We answer several questions of P. Erdos and R. Graham by showing that for any rational number r > 0, there exist integers n1, n2, ..., nk, k variable, where N < n1 < n2 < ... < nk < (e^r + o_r(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk.…
We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a…
An avoidance pattern where the letters within an occurrence of which are required to be adjacent is referred to as a subword. In this paper, we enumerate members of the set NC_n of non-crossing partitions of length n according to the number…
A pairing function for the non-negative integers is said to be binary perfect if the binary representation of the output is of length 2k or less whenever each input has length k or less. Pairing functions with square shells, such as the…
Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the…
Pseudorandom values are often generated as 64-bit binary words. These random words need to be converted into ranged values without statistical bias. We present an efficient algorithm to generate multiple independent uniformly-random bounded…
Recently Richard Stanley initiated a study of the distribution of the length as(w) of the longest alternating subsequence in a random permutation w from the symmetric group $S_n$. Among other things he found an explicit formula for the…
The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that…
The main result of the paper is the following. Let a non-degenerate distribution have finite moments $\mu_k$ of all orders $k=0,1,2,\ldots$. Then the sequence $\{\mu_k/k!, \; k=0,1,2,\ldots\}$ either contains infinitely many different terms…
An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has…
Let $(X_k)_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet $\mathcal{A}_m:=\{1,\dots,m\}$, and having respective probability mass function…
Run-length decoding is an operation on sequences in which a positive integer $a$ is replaced by a run(sequence of repeated elements) of length $a$. Iterated run-length decodings applied to sequences with alphabets consisting of pairs of…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…