15 research outputs found
Proofs of Conjectures about Pattern-Avoiding Linear Extensions
After fixing a canonical ordering (or labeling) of the elements of a finite
poset, one can associate each linear extension of the poset with a permutation.
Some recent papers consider specific families of posets and ask how many linear
extensions give rise to permutations that avoid certain patterns. We build off
of two of these papers. We first consider pattern avoidance in -ary heaps,
where we obtain a general result that proves a conjecture of Levin, Pudwell,
Riehl, and Sandberg in a special case. We then prove some conjectures that
Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding
linear extensions of rectangular posets.Comment: 9 pages, 2 figure
Pattern Avoidance in Poset Permutations
We extend the concept of pattern avoidance in permutations on a totally
ordered set to pattern avoidance in permutations on partially ordered sets. The
number of permutations on that avoid the pattern is denoted
. We extend a proof of Simion and Schmidt to show that for any poset , and we exactly classify the posets for which
equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and
improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated
author email addresse
On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions
Symbolic ultrametrics define edge-colored complete graphs K_n and yield a
simple tree representation of K_n. We discuss, under which conditions this idea
can be generalized to find a symbolic ultrametric that, in addition,
distinguishes between edges and non-edges of arbitrary graphs G=(V,E) and thus,
yielding a simple tree representation of G. We prove that such a symbolic
ultrametric can only be defined for G if and only if G is a so-called cograph.
A cograph is uniquely determined by a so-called cotree. As not all graphs are
cographs, we ask, furthermore, what is the minimum number of cotrees needed to
represent the topology of G. The latter problem is equivalent to find an
optimal cograph edge k-decomposition {E_1,...,E_k} of E so that each subgraph
(V,E_i) of G is a cograph. An upper bound for the integer k is derived and it
is shown that determining whether a graph has a cograph 2-decomposition, resp.,
2-partition is NP-complete
Locally Convex Words and Permutations
We introduce some new classes of words and permutations characterized by the
second difference condition , which we
call the -convexity condition. We demonstrate that for any sized alphabet
and convexity parameter , we may find a generating function which counts
-convex words of length . We also determine a formula for the number of
0-convex words on any fixed-size alphabet for sufficiently large by
exhibiting a connection to integer partitions. For permutations, we give an
explicit solution in the case and show that the number of 1-convex and
2-convex permutations of length are and ,
respectively, and use the transfer matrix method to give tight bounds on the
constants and . We also providing generating functions similar to
the the continued fraction generating functions studied by Odlyzko and Wilf in
the "coins in a fountain" problem.Comment: 20 pages, 4 figure
Pattern Avoidance in Reverse Double Lists
In this paper, we consider pattern avoidance in a subset of words on
called reverse double lists. In particular a reverse
double list is a word formed by concatenating a permutation with its reversal.
We enumerate reverse double lists avoiding any permutation pattern of length at
most 4 and completely determine the corresponding Wilf classes. For permutation
patterns of length 5 or more, we characterize when the number of
-avoiding reverse double lists on letters has polynomial growth. We
also determine the number of -avoiders of maximum length for any
positive integer .Comment: 24 pages, 5 figures, 4 table
Restricted 132-avoiding k-ary words, Chebyshev polynomials, and continued fractions
AbstractWe study generating functions for the number of n-long k-ary words that avoid both 132 and an arbitrary ℓ-ary pattern. In several interesting cases the generating function depends only on ℓ and is expressed via Chebyshev polynomials of the second kind and continued fractions
An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences
We introduce an algorithmic approach based on generating tree method for
enumerating the inversion sequences with various pattern-avoidance
restrictions. For a given set of patterns, we propose an algorithm that outputs
either an accurate description of the succession rules of the corresponding
generating tree or an ansatz. By using this approach, we determine the
generating trees for the pattern-classes ,
, , ,
and . Then we use the kernel method, obtain generating functions
of each class, and find enumerating formulas. Lin and Yan studied the
classification of the Wilf-equivalences for inversion sequences avoiding pairs
of length-three patterns and showed that there are 48 Wilf classes among 78
pairs. In this paper, we solve six open cases for such pattern classes.Comment: 20 pages, 2 figure
Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs
In this paper, we present a general methodology to solve a wide variety of classical lattice path counting problems in a uniform way. These counting problems are related to Dyck paths, Motzkin paths and some generalizations. The methodology uses weighted automata, equations of ordinary generating functions and continued fractions. This new methodology is called Counting Automata Methodology. It is a variation of the technique proposed by Rutten, which is called Coinductive Counting