12,766 research outputs found
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
Generalised Pattern Avoidance
Recently, Babson and Steingrimsson have introduced generalised permutation
patterns that allow the requirement that two adjacent letters in a pattern must
be adjacent in the permutation. We consider pattern avoidance for such
patterns, and give a complete solution for the number of permutations avoiding
any single pattern of length three with exactly one adjacent pair of letters.
We also give some results for the number of permutations avoiding two different
patterns. Relations are exhibited to several well studied combinatorial
structures, such as set partitions, Dyck paths, Motzkin paths, and involutions.
Furthermore, a new class of set partitions, called monotone partitions, is
defined and shown to be in one-to-one correspondence with non-overlapping
partitions
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
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
ANALYTIC AND TOPOLOGICAL COMBINATORICS OF PARTITION POSETS AND PERMUTATIONS
In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible
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