Related papers: A Spectral Approach to Consecutive Pattern-Avoidin…
We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting…
We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous…
A permutation $\pi$ is said to avoid a chain $(\sigma:\tau)$ of patterns if $\pi$ avoids $\sigma$ and $\pi^2$ avoids $\tau.$ In this paper, we define a notion of pattern avoidance for compositions of positive integers and use that idea to…
The number of 123-avoiding permutation on $\{1,2,\ldots,n\}$ with a fixed leading terms is counted by the ballot numbers. The same holds for $132$-avoiding permutations. These results were proved by Miner and Pak using the…
For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra…
The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given…
We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the…
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 prove that the number of permutations avoiding an arbitrary consecutive pattern of length m is asymptotically largest when the avoided pattern is 12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles a…
The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of…
In "Random complex fewnomials, I," B. Shiffman and S. Zelditch determine the limiting formula as N goes to infinity of the (normalized) expected distribution of complex zeros of a system of k random n-nomials in m variables where the…
There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to…
We consider the compound decision problem of estimating a vector of $n$ parameters, known up to a permutation, corresponding to $n$ independent observations, and discuss the difference between two symmetric classes of estimators. The first…
Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…
In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a…
Let $F \subset S_k$ be a finite set of permutations and let $C_n(F)$ denote the number of permutations $\sigma$ in $S_n$ avoiding the set of patterns $F$. The Noonan-Zeilberger conjecture states that the sequence ${C_n(F)}$ is P-recursive.…
Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the…
Given a set $\Pi$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(\Pi)$, the set of permutations of length at most $n$ avoiding the patterns in $\Pi$, in time $O(|S_{\le n - 1}(\Pi)| \cdot k +…
In recent work, Zeilberger and the author used a functional equations approach for enumerating permutations with r occurrences of the pattern 12...k. In particular, the approach yielded a polynomial-time enumeration algorithm for any fixed…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…