9,496 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
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
Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems
By algebraic group theory, there is a map from the semisimple conjugacy
classes of a finite group of Lie type to the conjugacy classes of the Weyl
group. Picking a semisimple class uniformly at random yields a probability
measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is
proved that this measure agrees with a second measure on conjugacy classes of
the Weyl group induced by a construction of Cellini using the affine Weyl
group. Formulas for Cellini's measure in type are found. This leads to new
models of card shuffling and has interesting combinatorial and number theoretic
consequences. An analysis of type C gives another solution to a problem of
Rogers in dynamical systems: the enumeration of unimodal permutations by cycle
structure. The proof uses the factorization theory of palindromic polynomials
over finite fields. Contact is made with symmetric function theory.Comment: One change: we fix a typo in definition of f(m,k,i,d) on page 1
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