23 research outputs found
Lehmer code transforms and Mahonian statistics on permutations
In 2000 Babson and Steingr{\'\i}msson introduced the notion of vincular
patterns in permutations. They shown that essentially all well-known Mahonian
permutation statistics can be written as combinations of such patterns. Also,
they proved and conjectured that other combinations of vincular patterns are
still Mahonian. These conjectures were proved later: by Foata and Zeilberger in
2001, and by Foata and Randrianarivony in 2006.
In this paper we give an alternative proof of some of these results. Our
approach is based on permutation codes which, like Lehmer's code, map
bijectively permutations onto subexcedant sequences. More precisely, we give
several code transforms (i.e., bijections between subexcedant sequences) which
when applied to Lehmer's code yield new permutation codes which count
occurrences of some vincular patterns
-Rook polynomials and matrices over finite fields
Connections between -rook polynomials and matrices over finite fields are
exploited to derive a new statistic for Garsia and Remmel's -hit polynomial.
Both this new statistic and another statistic for the -hit polynomial
recently introduced by Dworkin are shown to induce different multiset
Mahonian permutation statistics for any Ferrers board. In addition, for the
triangular boards they are shown to generate different families of
Euler-Mahonian statistics. For these boards the family includes Denert's
statistic , and gives a new proof of Foata and Zeilberger's Theorem that
is jointly distributed with . The family appears
to be new. A proof is also given that the -hit polynomials are symmetric and
unimodal
Combinatorial Statistics on Pattern-avoiding Permutations
The study of Mahonian statistics dated back to 1915 when MacMahon showed that
the major index and the inverse number have the same distribution on a set of
permutations with length n. Since then, many Mahonian statistics have been
discovered and much effort have been done to find the equidistribution between
two Mahonian statistics on permutations avoiding length-3 classical patterns.
In recent years, Amini and Do et al. have done extensive research with various
methods to prove the equidistributions, ranging from using generating
functions, Dyck paths, block decompositions, to bijections. In this thesis, we
will solve the conjectured equidistribution between bast and foze on Av(312)
using the bijection method, as well as refine two established results by Do et
al. with a combinatorial approach.Comment: 20 pages, 4 theorems, 13 lemmas, bachelor thesi
A permutation code preserving a double Eulerian bistatistic
Visontai conjectured in 2013 that the joint distribution of ascent and
distinct nonzero value numbers on the set of subexcedant sequences is the same
as that of descent and inverse descent numbers on the set of permutations. This
conjecture has been proved by Aas in 2014, and the generating function of the
corresponding bistatistics is the double Eulerian polynomial. Among the
techniques used by Aas are the M\"obius inversion formula and isomorphism of
labeled rooted trees. In this paper we define a permutation code (that is, a
bijection between permutations and subexcedant sequences) and show the more
general result that two -tuples of set-valued statistics on the set of
permutations and on the set of subexcedant sequences, respectively, are
equidistributed. In particular, these results give a bijective proof of
Visontai's conjecture
Generalizations of Permutation Statistics to Words and Labeled Forests
A classical result of MacMahon shows the equidistribution of the major index and inversion number over the symmetric groups. Since then, these statistics have been generalized in many ways, and many new permutation statistics have been defined, which are related to the major index and inversion number in may interesting ways. In this dissertation we study generalizations of some newer statistics over words and labeled forests.
Foata and Zeilberger defined the graphical major index, majU , and the graphical inversion index, invU , for words over the alphabet {1, . . . , n}. In this dissertation we define a graphical sorting index, sorU , which generalizes the sorting index of a permutation. We then characterize the graphs U for which sorU is equidistributed with invU and majU on a single rearrangement class.
Bj¨orner and Wachs defined a major index for labeled plane forests, and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, BT-max), (sor, Cyc), and (maj, CBT-max) are equidistributed. Our results extend the result of Bj¨orner and Wachs and generalize results for permutations.
Lastly, we study the descent polynomial of labeled forests. The descent polynomial for per-mutations is known to be log-concave and unimodal. In this dissertation we discuss what properties are preserved in the descent polynomial of labeled forests
Generating permutations with a given major index
In [S. Effler, F. Ruskey, A CAT algorithm for listing permutations with a
given number of inversions, {\it I.P.L.}, 86/2 (2003)] the authors give an
algorithm, which appears to be CAT, for generating permutations with a given
major index. In the present paper we give a new algorithm for generating a Gray
code for subexcedant sequences. We show that this algorithm is CAT and derive
it into a CAT generating algorithm for permutations with a given major index