Related papers: Omnibus Sequences, Coupon Collection, and Missing …
Segmental structure is a common pattern in many types of sequences such as phrases in human languages. In this paper, we present a probabilistic model for sequences via their segmentations. The probability of a segmented sequence is…
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this…
This paper is a primer on cryptographic accumulators and how to apply them practically. A cryptographic accumulator is a space- and time-efficient data structure used for set-membership tests. Since it is possible to represent any…
We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under…
A word $u$ is a subsequence of another word $w$ if $u$ can be obtained from $w$ by deleting some of its letters. The word $w$ with alph$(w)=\Sigma$ is called $k$-subsequence universal if the set of subsequences of length $k$ of $w$ contains…
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it…
Long sequence neural memory remains a challenging problem. RNNs and their variants suffer from vanishing gradients, and Transformers suffer from quadratic scaling. Furthermore, compressing long sequences into a finite fixed representation…
We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.
A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the…
The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…
Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the…
Consider the number of permutations in the symmetric group on n letters that contain c copies of a given pattern. As c varies (with n held fixed) these numbers form a sequence whose properties we study for the monotone patterns and the…
We study multiple simultaneous cut events for k-out-of-n:F and linear consecutive k-out-of-n:F systems in which each component has a constant failure probability. We list the multicuts of these systems and describe the structural…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…
These lecture notes cover basic automata-theoretic concepts and logical formalisms for the modeling and verification of concurrent and distributed systems. Many of these concepts naturally extend the classical automata and logics over…
We extend the Coupon Collector's Problem (CCP) and present a novel generalized model, referred as the k-LCCP problem, where one is interested in recovering a bipartite graph with a perfect matching, which represents the coupons and their…
A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms.…
The original knapsack problem is well known to be NP-complete. In a multidimensional version one have to decide whether a $p\in \N^k$ is in a sumset-sum of a set $X \subseteq \N^k$ or not. In this paper we are going to investigate a…
Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to…
This paper gives an algebraic proof of the correctness of Von Schelling formula for the probability of the coupon collector problem waiting time for non-uniform distributions and partial collections. It introduces a theorem on sums of…