Related papers: Inversions and Longest Increasing Subsequence for …
Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of…
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
In this paper, we investigate the reconstruction of permutations on {1, 2, ..., n} from betweenness constraints involving the minimum and the maximum element located between t and t+1, for all t=1, 2, ..., n-1. We propose two variants of…
Counting permutations of $[n]$ by the number of records, i.e. left-to-right maxima, is a classic problem in combinatorial enumeration. In the first volume of ``The Art of Computer Programming", Donald Knuth demonstrated its relevance for…
We introduce and analyze the $S_k$ shuffle on $N$ cards, a natural generalization of the celebrated random adjacent transposition shuffle. In the $S_k$ shuffle, we choose uniformly at random a block of $k$ consecutive cards, and shuffle…
In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_\theta^{(n)})_{\theta \ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the…
A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th…
A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th…
We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is…
We study two related probabilistic models of permutations and trees biased by their number of descents. Here, a descent in a permutation $\sigma$ is a pair of consecutive elements $\sigma(i), \sigma(i+1)$ such that $\sigma(i) >…
In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we…
We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain…
Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed…
The random transposition shuffle on repeated cards induces a Markov chain on the quotient space of arrangements with multiplicities, and is equivalent to the many-urn mean-field Bernoulli-Laplace model introduced by Scarabotti. Writing…
We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…
A superpermutation on $n$ symbols is a string that contains each of the $n!$ permutations of the $n$ symbols as a contiguous substring. The shortest superpermutation on $n$ symbols was conjectured to have length $\sum_{i=1}^n i!$. The…
In the top to random shuffle, the first a cards are removed from a deck of n cards 12 \cdots n and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element B_a, which we define…
Minimizers sampling is one of the most widely-used mechanisms for sampling strings [Roberts et al., Bioinformatics 2004]. Let $S=S[1]\ldots S[n]$ be a string over a totally ordered alphabet $\Sigma$. Further let $w\geq 2$ and $k\geq 1$ be…
The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on…
The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on…