Related papers: Block decomposition of permutations and Schur-posi…

The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied in this paper. We prove that the associated Chow quasi-symmetric generating function…

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux…

Bivariate generating functions for various subsets of the class of permutations containing no descending sequence of length three or more are determined. The notion of absolute indecomposability of a permutation is introduced, and used in…

Let $s$ denote West's stack-sorting map. A permutation is called $t-\textit{sorted}$ if it is of the form $s^t(\mu)$ for some permutation $\mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can…

Consider a random permutation of $kn$ objects that permutes $n$ disjoint blocks of size $k$ and then permutes elements within each block. Normalizing its cycle lengths by $kn$ gives a random partition of unity, and we derive the limit law…

Let $k$ be a nonnegative integer, and let $\alpha$ and $\beta$ be two permutations of $n$ symbols. We say that $\alpha$ and $\beta$ $k$-commute if $H(\alpha\beta, \beta\alpha)=k$, where $H$ denotes the Hamming metric between permutations.…

Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that…

A result of Foata and Schutzenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property:…

We prove a lower and an upper bound on the number of block moves necessary to sort a permutation. We put our results in contrast with existing results on sorting by block transpositions, and raise some open questions.

The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for…

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…

Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is…

For a set of permutations $S\subseteq S_n$, consider the quasisymmetric generating function $$Q(S): = \sum_{w\in S}F_{n, \mathrm{Des}(w)},$$ where $\mathrm{Des}(w) := \{i\mid w(i)> w(i+1)\}$ is the descent set of $w$ and $F_{n,…

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is…

A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating…

A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit…

Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and…

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1, \dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and…

We consider uniformly random set partitions of size $n$ with exactly $k$ blocks, and uniformly random permutations of size $n$ with exactly $k$ cycles, under the regime where $n-k \sim t\sqrt{n}$, $t>0$. In this regime, there is a simple…

We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic…