Related papers: Systematic Enumeration of Fundamental Quantities I…
When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\in\{H,T\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- and third-order linear recurrence relations and their generating functions to discuss the…
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
The goal of constraint-based sequence mining is to find sequences of symbols that are included in a large number of input sequences and that satisfy some constraints specified by the user. Many constraints have been proposed in the…
We study ranked enumeration of join-query results according to very general orders defined by selective dioids. Our main contribution is a framework for ranked enumeration over a class of dynamic programming problems that generalizes…
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
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…
The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern…
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli…
An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a…
Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives explicit formulas for Bernoulli numbers of even index. The formulas contain a remarkable sequence of determinants. The value of…
We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…
We study part sizes of supercritical locally restricted sequential structures. This extends previous results about locally restricted integer compositions and part sizes in smooth supercritical compositional structures. Applications are…
We study pairs and m--tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of…
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…
In this paper, we enumerate enumeration problems and algorithms. This survey is under construction. If you know some results not in this survey or there is anything wrong, please let me know.
We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for…
A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the…