Related papers: Non-standard binary representations and the Stern …
This paper introduces the sequence covering similarity, that we formally define for evaluating the similarity between a symbolic sequence (string) and a set of symbolic sequences (strings). From this covering similarity we derive a…
Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary…
It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth…
Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $\beta$ of $n$ in which the numbers from 1 to $n$ are colored…
Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers.…
Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$. Based on this, this…
It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices,…
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.
Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and…
We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short…
A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations…
We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b>0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and…
The article is devoted to a matrix method of comparative analysis of long nucleotide sequences by means of a presentation of each sequence in a form of three digital binary sequences. This method uses biochemical attributes of nucleotides…
In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices…
There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r+1. Motivated by work on rational numbers in the Calkin-Wilf tree, we classify partitions…
If the list of binary numbers is read by upward-sloping diagonals, the resulting ``sloping binary numbers'' 0, 11, 110, 101, 100, 1111, 1010, ... (or 0, 3, 6, 5, 4, 15, 10, ...) have some surprising properties. We give formulae for the n-th…
We extend our previous results on the number of integers which are values of some cyclotomic form of degree larger than a given value (see \cite{FW1}), to more general families of binary forms with integer coefficients. Our main ingredient…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
We prove that many sequences of positive numbers $(a_n)$ defined by finite linear difference equations $a_{n+k}=c_{k-1}a_{n+k-1}+...+c_0a_n$ with suitable non negative reals coefficients $c_i$ satisfy Bendford's Law on the first digit in…