Related papers: Omnibus Sequences, Coupon Collection, and Missing …

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…

Extractive summaries are usually presented as lists of sentences with no expected cohesion between them. In this paper, we aim to enforce cohesion whilst controlling for informativeness and redundancy in summaries, in cases where the input…

An ultraproduct can be a helpful organizing principle in presenting solutions of problems at many levels, as argued by Terence Tao. We apply it here to the solution of a calculus problem: every infinite sequence has a monotone infinite…

The problem of the universal compression of a sequence from a library of several small to moderate length sequences from similar context arises in many practical scenarios, such as the compression of the storage data and the Internet…

An emerging class of data systems partition their data and precompute approximate summaries (i.e., sketches and samples) for each segment to reduce query costs. They can then aggregate and combine the segment summaries to estimate results…

There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and…

Valleyless sequences of finite length $n$ and maximum entry $k$ occur in tree enumeration problems and provide an interesting correspondence between permutations and compositions. In this paper we introduce the notion of \emph {valleyless}…

An absent factor of a string $w$ is a string $u$ which does not occur as a contiguous substring (a.k.a. factor) inside $w$. We extend this well-studied notion and define absent subsequences: a string $u$ is an absent subsequence of a string…

This paper presents some first results on how to perform uniform random walks (where every trace has the same probability to occur) in very large models. The models considered here are described in a succinct way as a set of communicating…

An inversion sequence of length $n$ is an integer sequence $(a_1, \ldots, a_n)$ such that $0 \le a_i < i$ for all $i$. The study of pattern-avoiding inversion sequences was initiated in 2015 by Mansour and Shattuck and in 2016 by Corteel,…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…

A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and…

For every total recursive time bound $t$, a constant fraction of all compressible (low Kolmogorov complexity) strings is $t$-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of…

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…

We exhibit a one to one correspondence between some universal probabilistic properties of the ordering coordinate of one-dimensional Ising-like models and a class of continuous time random walks. This correspondence provides an new…

The fundamental question considered in algorithms on strings is that of indexing, that is, preprocessing a given string for specific queries. By now we have a number of efficient solutions for this problem when the queries ask for an exact…

In this note we show that if $(u_n)_{n\geqslant 1}$ is a simple linearly recurrent sequence of integers whose minimal recurrence of order $k$ involves only positive coefficients that has positive initial terms, then $(Mu_{n^s})_{n\geqslant…

The sequence of partial sums of Fibonacci numbers, beginning with $2$, $4$, $7$, $12$, $20$, $33,\dots$, has several combinatorial interpretations (OEIS A000071). For instance, the $n$-th term in this sequence is the number of length-$n$…

We consider the issue of direct access to any letter of a sequence encoded with a variable length code and stored in the computer's memory, which is a special case of the random access problem to compressed memory. The characteristics…

The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…