Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of 0021-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern-restricted permutations. The objective of this paper is not only a confirmation of Hong and Li’s conjecture and Burstein’s first conjecture but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.

The higher-dimensional Thompson groups $nV$, for $n \geqslant 2$, were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra 284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

We investigate random minimal factorizations of the n-cycle, that is, factorizations of the permutation $(1 \, 2 \cdots n)$ into a product of cycles $\tau_1, \ldots, \tau_k$ whose lengths $\ell(\tau_1), \ldots, \ell(\tau_k)$ satisfy the minimality condition $\sum_{i=1}^k(\ell(\tau_i)-1)=n-1$ . By associating to a cycle of the factorization a black polygon inscribed in the unit disk, and reading the cycles one after another, we code a minimal factorization by a process of colored laminations of the disk. These new objects are compact subsets made of red noncrossing chords delimiting faces that are either black or white. Our main result is the convergence of this process as $n \rightarrow \infty$ , when the factorization is randomly chosen according to Boltzmann weights in the domain of attraction of an $\alpha$ -stable law, for some $\alpha \in (1,2]$ . The limiting process interpolates between the unit circle and a colored version of Kortchemski’s $\alpha$ -stable lamination. Our principal tool in the study of this process is a new bijection between minimal factorizations and a model of size-conditioned labeled random trees whose vertices are colored black or white, as well as the investigation of the asymptotic properties of these trees.

Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in \mathbb {Z}$ be a primitive root modulo p. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo p in the interval $(0,p)$ . Then clearly the sequence $a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$ is a permutation of the sequence $g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$ . We determine the sign of this permutation.

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

In this paper we present a combinatorial proof of an identity involving the two kinds of Stirling numbers and the numbers of permutations with prescribed numbers of excedances and cycles. Several recurrence relations related to the numbers of excedances, fixed points and cycles are also obtained.

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

The geographical distance between conspecific host populations is no doubt a key determinant of the likelihood that exchanges of parasite species occur between these populations. This variable must therefore be taken into account in studies that compare parasite species richness or similarity among host populations. This paper presents a multivariate approach, based on the permutation of matrices, that allows all pairwise geographical distances between host populations to be included as independent variables. The method is illustrated with 3 separate data sets on parasite communities of conspecific fish from different lakes. In 2 of 3 cases, geographical distances among lakes had a significant influence on the similarity of their parasite communities. The effect of geographical distance on species richness of parasite communities also proved important in 2 of the 3 case studies. These examples demonstrate the pervasive influence of distances among host populations on their parasite communities, and the need to properly control for them in statistical analyses.

Define a random variable

We refer to Yn as a sustained maximum. Under the assumption that the Xi's are i.i.d. the exact distribution of Yn is obtained. Under certain conditions on the underlying distribution F we obtain weak limit results for the Yn as well. Also a combinatorial extreme-value problem is solved.