Related papers: Longest subsequence for certain repeated up/down p…
Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected…
We prove a conjecture of Drew Armstrong on the average maximal length of $k$-alternating subsequence of permutations. The $k=1$ case is a well-known result of Richard Stanley.
Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the…
Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in S_n which contain a given permutation \tau in S_k as a subsequence; this number depends on the patterns of the first j values of \tau for 1<=j<=k. We…
The longest square subsequence (LSS) problem consists of computing a longest subsequence of a given string $S$ that is a square, i.e., a longest subsequence of form $XX$ appearing in $S$. It is known that an LSS of a string $S$ of length…
In this paper, we examine the asymptotic behavior of the longest increasing subsequence (LIS) in a uniformly random permutation of $n$ elements. We rely on the Robinson--Schensted--Knuth correspondence, Young tableaux, and key classical…
A permutation $\sigma \in S_n$ is a $k$-superpattern (or $k$-universal) if it contains each $\tau \in S_k$ as a pattern. This notion of "superpatterns" can be generalized to words on smaller alphabets, and several questions about…
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…
We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{\alpha_1, \alpha_2, \cdots, \alpha_m\}$, while the…
The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of…
The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits sqrt[n](|S_n(tau)|) and further works suggest asymptotic ordering of layered versus monotone patterns. Yet,…
A repetition free Longest Common Subsequence (LCS) of two sequences x and y is an LCS of x and y where each symbol may appear at most once. Let R denote the length of a repetition free LCS of two sequences of n symbols each one chosen…
We study the shape of the Young diagram \lambda associated via the Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other…
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.…
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] , \dots w[i_{|u|}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq |w|$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if every word…
We study numerically the distributions of the length $L$ of the longest increasing subsequence (LIS) for the two cases of random permutations and of one-dimensional random walks. Using sophisticated large-deviation algorithms, we are able…
For $\tau\in S_3$, let $\mu_n^{\tau}$ denote the uniformly random probability measure on the set of $\tau$-avoiding permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by…
We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern…
Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the…
In this note, we consider the problem of finding a step-by-step transformation between two longest increasing subsequences in a sequence, namely Longest Increasing Subsequence Reconfiguration. We give a polynomial-time algorithm for…