Related papers: Optimal Regular Expressions for Permutations

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…

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…

The extension complexity of a polytope $P$ is the smallest integer $k$ such that $P$ is the projection of a polytope $Q$ with $k$ facets. We study the extension complexity of $n$-gons in the plane. First, we give a new proof that the…

We prove that the minimal size $M(\pi_n)$ of a maximal matching in the permutahedron $\pi_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(\pi_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron.…

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…

The downward and upward closures of a regular language $L$ are obtained by collecting all the subwords and superwords of its elements, respectively. The downward and upward interiors of $L$ are obtained dually by collecting words having all…

We improve the best known upper bound on the length of the shortest reset words of synchronizing automata. The new bound is slightly better than $114 n^3 / 685 + O(n^2)$. The \v{C}ern\'y conjecture states that $(n-1)^2$ is an upper bound.…

We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations…

Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming…

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square…

We study 4 problems in string matching, namely, regular expression matching, approximate regular expression matching, string edit distance, and subsequence indexing, on a standard word RAM model of computation that allows logarithmic-sized…

A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1…

A reduced word of a permutation $w$ is a minimal length expression of $w$ as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove…

We study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the…

A $k$-universal permutation, or $k$-superpermutation, is a permutation that contains all permutations of length $k$ as patterns. The problem of finding the minimum length of a $k$-superpermutation has recently received significant attention…

The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters…

In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the…

The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding…

We address a question and a conjecture on the expected length of the longest common subsequences of two i.i.d.$\ $random permutations of $[n]:=\{1,2,...,n\}$. The question is resolved by showing that the minimal expectation is not attained…

We prove two results about width of words in $SL_n(\mathbb{Z})$. The first is that, for every $n \geq 3$, there is a constant $C(n)$ such that the width of any word in $SL_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for…