65 research outputs found
Counting Self-Dual Interval Orders
In this paper, we present a new method to derive formulas for the generating
functions of interval orders, counted with respect to their size, magnitude,
and number of minimal and maximal elements. Our method allows us not only to
generalize previous results on refined enumeration of general interval orders,
but also to enumerate self-dual interval orders with respect to analogous
statistics.
Using the newly derived generating function formulas, we are able to prove a
bijective relationship between self-dual interval orders and upper-triangular
matrices with no zero rows. Previously, a similar bijective relationship has
been established between general interval orders and upper-triangular matrices
with no zero rows and columns.Comment: 20 page
Splittability and 1-amalgamability of permutation classes
A permutation class is splittable if it is contained in a merge of two of
its proper subclasses, and it is 1-amalgamable if given two permutations
and in , each with a marked element, we can find a
permutation in containing both and such that the two
marked elements coincide. It was previously shown that unsplittability implies
1-amalgamability. We prove that unsplittability and 1-amalgamability are not
equivalent properties of permutation classes by showing that the class
is both splittable and 1-amalgamable. Our construction is
based on the concept of LR-inflations, which we introduce here and which may be
of independent interest.Comment: 17 pages, 7 figure
On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admit an
intersection representation by L-shaped curves, where additionally the topmost
points of each curve are assumed to belong to a common horizontal line. We
prove that Grounded-L graphs admit an equivalent characterisation in terms of
vertex ordering with forbidden patterns.
We also compare this class to related intersection classes, such as the
grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance
graphs), or the outer-1-string graphs. We give constructions showing that these
classes are all distinct and satisfy only trivial or previously known
inclusions.Comment: 16 pages, 6 figure
On Multiple Pattern Avoiding Set Partitions
We study classes of set partitions determined by the avoidance of multiple
patterns, applying a natural notion of partition containment that has been
introduced by Sagan. We say that two sets S and T of patterns are equivalent if
for each n, the number of partitions of size n avoiding all the members of S is
the same as the number of those that avoid all the members of T.
Our goal is to classify the equivalence classes among two-element pattern
sets of several general types. First, we focus on pairs of patterns
{\sigma,\tau}, where \sigma\ is a pattern of size three with at least two
distinct symbols and \tau\ is an arbitrary pattern of size k that avoids
\sigma. We show that pattern-pairs of this type determine a small number of
equivalence classes; in particular, the classes have on average exponential
size in k. We provide a (sub-exponential) upper bound for the number of
equivalence classes, and provide an explicit formula for the generating
function of all such avoidance classes, showing that in all cases this
generating function is rational.
Next, we study partitions avoiding a pair of patterns of the form
{1212,\tau}, where \tau\ is an arbitrary pattern. Note that partitions avoiding
1212 are exactly the non-crossing partitions. We provide several general
equivalence criteria for pattern pairs of this type, and show that these
criteria account for all the equivalences observed when \tau\ has size at most
six.
In the last part of the paper, we perform a full classification of the
equivalence classes of all the pairs {\sigma,\tau}, where \sigma\ and \tau\
have size four.Comment: 37 pages. Corrected a typ
Dyck paths and pattern-avoiding matchings
How many matchings on the vertex set V={1,2,...,2n} avoid a given
configuration of three edges? Chen, Deng and Du have shown that the number of
matchings that avoid three nesting edges is equal to the number of matchings
avoiding three pairwise crossing edges. In this paper, we consider other
forbidden configurations of size three. We present a bijection between
matchings avoiding three crossing edges and matchings avoiding an edge nested
below two crossing edges. This bijection uses non-crossing pairs of Dyck paths
of length 2n as an intermediate step.
Apart from that, we give a bijection that maps matchings avoiding two nested
edges crossed by a third edge onto the matchings avoiding all configurations
from an infinite family, which contains the configuration consisting of three
crossing edges. We use this bijection to show that for matchings of size n>3,
it is easier to avoid three crossing edges than to avoid two nested edges
crossed by a third edge.
In this updated version of this paper, we add new references to papers that
have obtained analogous results in a different context.Comment: 18 pages, 4 figures, important references adde
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