Related papers: Statistics on clusters and $r$-Stirling permutatio…
A $k$-Stirling permutation of order $n$ is said to be "flattened" if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a…
It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$…
Because of its mathematical tractability, the Gaussian mixture model holds a special place in the literature for clustering and classification. For all its benefits, however, the Gaussian mixture model poses problems when the data is skewed…
We improve current instability-based methods for the selection of the number of clusters $k$ in cluster analysis by developing a normalized cluster instability measure that corrects for the distribution of cluster sizes, a previously…
A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates…
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right…
The Euclidean k-means problem is arguably the most widely-studied clustering problem in machine learning. While the k-means objective is NP-hard in the worst-case, practitioners have enjoyed remarkable success in applying heuristics like…
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist,…
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist,…
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in…
A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle…
Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic…
The finding of multiple stellar populations (MP), defined by patterns in the stellar element abundances, is nowadays considered a distinctive feature of globular clusters. However, while data availability and quality improved in last…
We consider the well-studied pattern counting problem: given a permutation $\pi \in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $\tau \in \mathbb{S}_k$ in $\pi$. Our first result…
We consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail…
Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics…
Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any…
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain…
Typically clustering algorithms provide clustering solutions with prespecified number of clusters. The lack of a priori knowledge on the true number of underlying clusters in the dataset makes it important to have a metric to compare the…
Most Mahonian statistics can be expressed as a linear combination of vincular patterns. This is not only true with statistics on the permutation set, but it can also be applied for statistics on the permutation with repetition set. By…