138 research outputs found
Permutation Statistics and Pattern Avoidance in Involutions
Dokos et. al. studied the distribution of two statistics over permutations
of that avoid one or more length three
patterns. A permutation contains a pattern
if has a subsequence of length whose
letters are in the same relative order as . This paper is a comprehensive
study of the same two statistics, number of inversions and major index, over
involutions that
avoid one or more length three patterns. The equalities between the generating
functions are consequently determined via symmetries and we conjecture this
happens for longer patterns as well. We describe the generating functions for
each set of patterns including the fixed-point-free case, for
all Notating as the generating function for the
major index over the avoidance class of involutions associated to we
particularly present an independent determination that is
the -analogue for the central binomial coefficient that first appeared in a
paper by Barnebei, Bonetti, Elizalde and Silimbani. A shorter proof is
presented that establishes a connection to core, a central topic in poset
theory. We also prove that
and that the
same symmetry holds for the larger class of permutations conjecturing that the
same equality is true for involutions and permutations given any pair of
patterns of the form and ,
Permutation patterns and statistics
Let S_n denote the symmetric group of all permutations of the set {1, 2,
...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we
let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of
Pi in the sense of pattern avoidance. One of the celebrated notions in pattern
theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if
#Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage
proposed studying a q-analogue of this concept defined as follows. Suppose
st:S->N is a permutation statistic where N represents the nonnegative integers.
Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in
Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if
F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth
study of this concept for the inv and maj statistics. In particular, we
determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This
leads us to consider various q-analogues of the Catalan numbers, Fibonacci
numbers, triangular numbers, and powers of two. Our proof techniques use
lattice paths, integer partitions, and Foata's fundamental bijection. We also
answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of
the conjectures have been prove
Block patterns in Stirling permutations
We introduce and study a new notion of patterns in Stirling and -Stirling
permutations, which we call block patterns. We prove a general result which
allows us to compute generating functions for the occurrences of various block
patterns in terms of generating functions for the occurrences of patterns in
permutations. This result yields a number of applications involving, among
other things, Wilf equivalence of block patterns and a new interpretation of
Bessel polynomials. We also show how to interpret our results for a certain
class of labeled trees, which are in bijection with Stirling permutations
Pattern avoidance in matchings and partitions
Extending the notion of pattern avoidance in permutations, we study matchings
and set partitions whose arc diagram representation avoids a given
configuration of three arcs. These configurations, which generalize 3-crossings
and 3-nestings, have an interpretation, in the case of matchings, in terms of
patterns in full rook placements on Ferrers boards.
We enumerate 312-avoiding matchings and partitions, obtaining algebraic
generating functions, in contrast with the known D-finite generating functions
for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a
more direct proof of a formula of B\'ona for the number of 1342-avoiding
permutations. Additionally, we give a bijection proving the
shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies
existing proofs by Backelin--West--Xin and Jel\'{\i}nek, and provides an
extension of work of Gouyou-Beauchamps for matchings with fixed points.
Finally, we classify pairs of patterns of length 3 according to
shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair
in most of the resulting equivalence classes.Comment: 34 pages, 12 Figures, 5 Table
Pattern avoidance for set partitions \`a la Klazar
In 2000 Klazar introduced a new notion of pattern avoidance in the context of
set partitions of . The purpose of the present paper is to
undertake a study of the concept of Wilf-equivalence based on Klazar's notion.
We determine all Wilf-equivalences for partitions with exactly two blocks, one
of which is a singleton block, and we conjecture that, for , these are
all the Wilf-equivalences except for those arising from complementation. If
is a partition of and denotes the set of all
partitions of that avoid , we establish inequalities between
and for several choices of and
, and we prove that if is the partition of with only one
block, then and all partitions
of with exactly two blocks. We conjecture that this result holds
for all partitions of . Finally, we enumerate for
all partitions of .Comment: 21 page
Egge triples and unbalanced Wilf-equivalence
Egge conjectured that permutations avoiding the set of patterns
, where ,
are enumerated by the large Schr\"oder numbers. Consequently,
with as above is Wilf-equivalent to the set of
patterns . Burstein and Pantone proved the case of
. We prove the remaining four cases. As a byproduct of our proof,
we also enumerate the case .Comment: 20 pages, 6 figures (published version
A general theory of Wilf-equivalence for Catalan structures
The existence of apparently coincidental equalities (also called
Wilf-equivalences) between the enumeration sequences, or generating functions,
of various hereditary classes of combinatorial structures has attracted
significant interest. We investigate such coincidences among non-crossing
matchings and a variety of other Catalan structures including Dyck paths,
231-avoiding permutations and plane forests. In particular we consider
principal classes defined by not containing an occurrence of a single given
structure. An easily computed equivalence relation among structures is
described such that if two structures are equivalent then the associated
principal classes have the same enumeration sequence. We give an asymptotic
estimate of the number of equivalence classes of this relation among structures
of a given size and show that it is exponentially smaller than the
corresponding Catalan number. In other words these "coincidental" equalities
are in fact very common among principal classes. Our results also allow us to
prove, in a unified and bijective manner, several known Wilf-equivalences from
the literature.Comment: 24page
Matchings Avoiding Partial Patterns
We show that matchings avoiding certain partial patterns are counted by the
3-Catalan numbers. We give a characterization of 12312-avoiding matchings in
terms of restrictions on the corresponding oscillating tableaux. We also find a
bijection between Schr\"oder paths without peaks at level one and matchings
avoiding both patterns 12312 and 121323. Such objects are counted by the
super-Catalan numbers or the little Schr\"{o}der numbers. A refinement of the
super-Catalan numbers is obtained by fixing the number of crossings in the
matchings. In the sense of Wilf-equivalence, we find that the patterns 12132,
12123, 12321, 12231, 12213 are equivalent to 12312.Comment: 17 pages, 7 figure
Refined Wilf-equivalences by Comtet statistics
We launch a systematic study of the refined Wilf-equivalences by the
statistics and , where and
are the number of components and the length of the initial
ascending run of a permutation , respectively. As Comtet was the first one
to consider the statistic in his book {\em Analyse
combinatoire}, any statistic equidistributed with over a class
of permutations is called by us a {\em Comtet statistic} over such class. This
work is motivated by a triple equidistribution result of Rubey on
-avoiding permutations, and a recent result of the first and third authors
that is a Comtet statistic over separable permutations. Some
highlights of our results are:
(1) Bijective proofs of the symmetry of the double Comtet distribution
over several Catalan and Schr\"oder classes,
preserving the values of the left-to-right maxima.
(2) A complete classification of - and
-Wilf-equivalences for length patterns and pairs of length
patterns. Calculations of the
generating functions over these pattern avoiding classes and separable
permutations.
(3) A further refinement by the Comtet statistic , of Wang's
recent descent-double descent-Wilf equivalence between separable permutations
and -avoiding permutations.Comment: 39 pages, 2 tables, 2 figures. Comments are welcom
-arrangements, statistics and patterns
The -arrangements are permutations whose fixed points are -colored. We
prove enumerative results related to statistics and patterns on
-arrangements, confirming several conjectures by Blitvi\'c and
Steingr\'imsson. In particular, one of their conjectures regarding the
equdistribution of the number of descents over the derangement form and the
permutation form of -arrangements is strengthened in two interesting ways.
Moreover, as one application of the so-called Decrease Value Theorem, we
calculate the generating function for a symmetric pair of Eulerian statistics
over permutations arising in our study.Comment: 25 pages, 1 figure and 1 tabl
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