The sorting index

We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian statistic, which we call the sorting index. The sorting index of a permutation and its type B and type D analogues have natural combinatorial descriptions which we describe in detail.Comment: 14 pages, minor changes, new references adde

The Sorting Index and Permutation Codes

In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called the sorting index. Petersen proved that the pairs of statistics $(sor,cyc)$ and $(inv,rl\textrm{-}min)$ have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, and we show that the bijection of Foata and Han serves the purpose of mapping $(inv,rl\textrm{-}min)$ to $(sor,cyc)$. We also give a type $B$ analogue of the Foata-Han bijection, and we derive the quidistribution of $(inv_B,{\rm Lmap_B},{\rm Rmil_B})$ and $(sor_B,{\rm Lmap_B},{\rm Cyc_B})$ over signed permutations. So we get a combinatorial interpretation of Petersen's equidistribution of $(inv_B,nmin_B)$ and $(sor_B,l_B')$. Moreover, we show that the six pairs of set-valued statistics $\rm (Cyc_B,Rmil_B)$, $\rm(Cyc_B,Lmap_B)$, $\rm(Rmil_B,Lmap_B)$, $\rm(Lmap_B,Rmil_B)$, $\rm(Lmap_B,Cyc_B)$ and $\rm(Rmil_B,Cyc_B)$ are equidistributed over signed permutations. For Coxeter groups of type $D$, Petersen showed that the two statistics $inv_D$ and $sor_D$ are equidistributed. We introduce two statistics $nmin_D$ and $\tilde{l}_D'$ for elements of $D_n$ and we prove that the two pairs of statistics $(inv_D,nmin_D)$ and $(sor_D,\tilde{l}_D')$ are equidistributed.Comment: 25 page

Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics (\sor, \cyc) and (\inv, \mathrm{rlmin}) are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sorting index for bicolored matchings and use it to show analogous equidistribution results for restricted permutations of type $B_n$ and $D_n$.Comment: 23 page

Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters

Combinatorial channels from partially ordered sets

A central task of coding theory is the design of schemes to reliably transmit data though space, via communication systems, or through time, via storage systems. Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets. We represent adversarial error models as combinatorial channels, form combinatorial channels from posets, identify a structural property of posets that leads to families of channels with the same codes, and bound the size of codes by optimizing over a family of equivalent channels. A large number of previously studied coding problems that fit into this framework. This leads to a new upper bound on the size of s-deletion correcting codes. We use a linear programming framework to obtain sphere-packing upper bounds when there is little underlying symmetry in the coding problem. Finally, we introduce and investigate a strong notion of poset homomorphism: locally bijective cover preserving maps. We look for maps of this type to and from the subsequence partial order on q-ary strings