504 research outputs found
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
Introduction to Partially Ordered Patterns
We review selected known results on partially ordered patterns (POPs) that
include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified)
maxima and minima) in permutations, the Horse permutations and others. We
provide several (new) results on a class of POPs built on an arbitrary flat
poset, obtaining, as corollaries, the bivariate generating function for the
distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and
Pell numbers, as well as generalizations of few results in the literature
including the descent distribution. Moreover, we discuss q-analogue for a
result on non-overlapping segmented POPs. Finally, we suggest several open
problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea
Descent c-Wilf Equivalence
Let denote the symmetric group. For any , we let
denote the number of descents of ,
denote the number of inversions of , and
denote the number of left-to-right minima of .
For any sequence of statistics on
permutations, we say two permutations and in are
-c-Wilf equivalent if the generating
function of over all permutations which
have no consecutive occurrences of equals the generating function of
over all permutations which have no
consecutive occurrences of . We give many examples of pairs of
permutations and in which are -c-Wilf
equivalent, -c-Wilf equivalent, and
-c-Wilf equivalent. For example, we
will show that if and are minimally overlapping permutations
in which start with 1 and end with the same element and
and , then and are
-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur
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
Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari
order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths
or (m+1)-ary trees. On another hand, the Tamari order is related to the product
in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new
combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is
described by the m-Tamari lattices.
In the same way as planar binary trees can be interpreted as sylvester
classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what
we call m-permutations. These objects are no longer in bijection with
decreasing (m+1)-ary trees, and a finer congruence, called metasylvester,
allows us to build Hopf algebras based on these decreasing trees. At the
opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of
graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions
and quasi-symmetric functions in a natural way. Finally, the algebras of packed
words and parking functions also admit such m-analogues, and we present their
subalgebras and quotients induced by the various congruences.Comment: 51 page
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