73 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
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
Two Reflected Gray Code based orders on some restricted growth sequences
We consider two order relations: that induced by the m-ary reflected Gray
code and a suffix partitioned variation of it. We show that both of them when
applied to some sets of restricted growth sequences still yield Gray codes.
These sets of sequences are: subexcedant or ascent sequences, restricted growth
functions, and staircase words. In each case we give efficient exhaustive
generating algorithms and compare the obtained results
Mahonian STAT on words
In 2000, Babson and Steingr\'imsson introduced the notion of what is now
known as a permutation vincular pattern, and based on it they re-defined known
Mahonian statistics and introduced new ones, proving or conjecturing their
Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by
Foata and Randrianarivony in 2006.
In 2010, Burstein refined some of these results by giving a bijection between
permutations with a fixed value for the major index and those with the same
value for STAT, where STAT is one of the statistics defined and proved to be
Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other
statistics are preserved as well by Burstein's bijection.
At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in
2010, Burstein asked whether his bijection has other interesting properties. In
this paper, we not only show that Burstein's bijection preserves the Eulerian
statistic ides, but also use this fact, along with the bijection itself, to
prove Mahonity of the statistic STAT on words we introduce in this paper. The
words statistic STAT introduced by us here addresses a natural question on
existence of a Mahonian words analogue of STAT on permutations. While proving
Mahonity of our STAT on words, we prove a more general joint equidistribution
result involving two six-tuples of statistics on (dense) words, where
Burstein's bijection plays an important role
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
Gray code order for Lyndon words
International audienceAt the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order
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