We introduce and study new refinements of inversion statistics for permutations, such as k-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for k-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we introduce, such as (\leqk)-step inversions and (k1,k2)-step inversions (that fix the value separation as well as the position). We connect our refinements to other work, such as inversion tops that are 0 modulo a fixed integer d, left boundary sums of paths, and marked meshed patterns. Finally, we use non-inversion sums to show that for every number n > 34, there is a permutation such that the dot product of that permutation and the identity permutation (of the same length) is n.
A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. In this article, we obtained a formula in close form for the multivariate generating function of {A(n,d,j)}, which denote the number of permutations of length n with d descents and j as the first letter. Besides, by a series of calculations with generatingfunctionology, we confirm a recent conjecture of Wang and Zhang for ballot permutations.
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where $1 \le i \textless{} j \le n$, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and…
We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${\boldsymbolσ}$ is a permutation chosen uniformly at random from the set of all permutations of $[n]$ with exactly $m=m(n)\ll n^2$ inversions. If $i<j$ are chosen uniformly at random from $[n]$, then ${\boldsymbolσ}(i)<{\boldsymbolσ}(j)$ asymptotically almost surely. However, if $i$ and $j$ are chosen so that $j-i\ll m/n$, and $m \ll n^2/\log^2 n$, then $\lim_{n\to\infty}\mathbb{P}\big[{\boldsymbolσ}(i)<{\boldsymbolσ}(j)\big]=\frac{1}{2}$. Moreover, if $k=k(n)\ll \sqrt{m/n}$, then t…
Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $Θ:S_n(321)\rightarrow S_…
We consider the joint distribution of the area and perimeter statistics on the set I_n of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distribution. Explicit formulas are found in some special cases as are expressions for the totals of the respective statistics on I_n. A similar treatment is provided for the joint distribution on I_n for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of…
Recently, the study of patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by {\em Catalan}, {\em Large Schröder}, {\em Baxter} and {\em Euler} numbers. One of the two highlights of our results is a fascinating bijection between $000$-avoiding inversion sequences and Simsun permutations, which together with Foata'…
The Robinson-Schensted correspondence can be viewed as a map from permutations to partitions. In this work, we study the number of inversions of permutations corresponding to a fixed partition $λ$ under this map. Hohlweg characterized permutations having shape $λ$ with the minimum number of inversions. Here, we give the first results in this direction for higher numbers of inversions. We give explicit conjectures for both the structure and the number of permutations associated to $λ$ where the extra number of inversions is less than the length of the smallest column of $λ$. We prove the result…
We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics, providing a framework in which the most well-known equidistributivity results for classical statistics are paralleled. We explore additional relationships among the statistics providing specific formulas in certain special cases. Moreover, we explore the behavior of these width-$k$ statistics in the context of pattern avoidance.
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 per…
For each permutation $w$, we can construct a collection of hyperplanes $\mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman, Linusson, Shareshian and Sjöstrand that the number of regions of $\mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $\mathcal{A}_w$ is greater than…
We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a metric on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this metric extends naturally to balanced tableaux, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.
We provide two explicit bijections demonstrating that, among permutations, the number of invisible inversions is equidistributed with the number of occurrences of the vincular pattern 13-2 after sorting the set of runs.
We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of $[n] \times [n]$ to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over …
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 maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijection…
We present a short proof of MacMahon's classic result that the number of permutations with $k$ inversions equals the number whose major index (sum of positions at which descents occur) is $k$
We launch a systematic study of the refined Wilf-equivalences by the statistics $\mathsf{comp}$ and $\mathsf{iar}$, where $\mathsf{comp}(π)$ and $\mathsf{iar}(π)$ are the number of components and the length of the initial ascending run of a permutation $π$, respectively. As Comtet was the first one to consider the statistic $\mathsf{comp}$ in his book {\em Analyse combinatoire}, any statistic equidistributed with $\mathsf{comp}$ over a class of permutations is called by us a {\em Comtet statistic} over such class. This work is motivated by a triple equidistribution result of Rubey on $321$-avo…
Consider the regular representation of the sum over all permutations weighted by the sum of their descent, inversion, and fixed point multinomials. We compute the spectrum and the multiplicities of its elements of that matrix. Note that those multinomial statistics allow to apply the result on several permutation statistics like the number of fixed points, of descents, of inversions, and the major index at the same time.
Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableau…
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 turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the $n$-th large Schröde…
If $α\in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $α$ is $Φ(α) = \{(i, j) \, | \, 1 \leq i < j \leq n, α(i) > α(j)\}$. We describe all $r$-tuples $α_1, α_2, \ldots, α_r \in S_n$ such that $Δ_n^+ = \{(i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $Φ(α_1), Φ(α_2), \ldots, Φ(α_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous problems for the Weyl groups of root systems of types $B$, $C$ and $D$ provid…
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the inversion words of permutations and the "usual" representation of descending plane partitions as families of non--intersec\-ting lattice paths.
