57 research outputs found
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 ,
called the sorting index. Petersen proved that the pairs of statistics
and 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
to . We also give a type analogue of the
Foata-Han bijection, and we derive the quidistribution of and over signed
permutations. So we get a combinatorial interpretation of Petersen's
equidistribution of and . Moreover, we show that
the six pairs of set-valued statistics ,
, , ,
and are equidistributed over signed
permutations. For Coxeter groups of type , Petersen showed that the two
statistics and are equidistributed. We introduce two statistics
and for elements of and we prove that the two
pairs of statistics and 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 non-atacking rooks on a
Ferrers board with rows and 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 and .Comment: 23 page
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
- β¦