25,330 research outputs found
width-k Eulerian polynomials of type A and B and its Gamma-positivity
We define some generalizations of the classical descent and inversion
statistics on signed permutations that arise from the work of Sack and
Ulfarsson [20] and called after width-k descents and width-k inversionsof type
A in Davis's work [8]. Using the aforementioned new statistics, we derive some
new generalizations of Eulerian polynomials of type A, B and D. It should also
be noticed that we establish the Gamma-positivity of the "width-k" Eulerian
polynomials and we give a combinatorial interpretation of finite sequences
associated to these new polynomials using quasisymmetric functions and
P-partition in Petersen's work [18].Comment: 28 page
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
Statistics on ordered partitions of sets
We introduce several statistics on ordered partitions of sets, that is, set
partitions where the blocks are permuted arbitrarily. The distribution of these
statistics is closely related to the q-Stirling numbers of the second kind.
Some of the statistics are generalizations of known statistics on set
partitions, but others are entirely new. All the new ones are sums of two
statistics, inspired by statistics on permutations, where one of the two
statistics is based on a certain partial ordering of the blocks of a partition.Comment: Added a Prologue, as this paper is soon to be published in a journa
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
Asymptotic behavior of some statistics in Ewens random permutations
The purpose of this article is to present a general method to find limiting
laws for some renormalized statistics on random permutations. The model
considered here is Ewens sampling model, which generalizes uniform random
permutations. We describe the asymptotic behavior of a large family of
statistics, including the number of occurrences of any given dashed pattern.
Our approach is based on the method of moments and relies on the following
intuition: two events involving the images of different integers are almost
independent.Comment: 32 pages: final version for EJP, produced by the author. An extended
abstract of 12 pages, published in the proceedings of AofA 2012, is also
available as version
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