15,026 research outputs found
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
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
Ascent Sequences Avoiding Pairs of Patterns
Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length n role= presentation style= display: inline; font-size: 11.2px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; position: relative; \u3enn avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound
Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers
This is the first of three papers that develop structures which are counted
by a "parabolic" generalization of Catalan numbers. Fix a subset R of
{1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are
determined by R. These are the "inverses" of (parabolic) multipermutations
whose multiplicities are determined by R. The standard forms of the ordered
partitions are refered to as "R-permutations". The notion of 312-avoidance is
extended from permutations to R-permutations. Let lambda be a partition of N
such that the set of column lengths in its shape is R or R union {n}. Fix an
R-permutation pi. The type A Demazure character (key polynomial) in x_1, ..,
x_n that is indexed by lambda and pi can be described as the sum of the weight
monomials for some of the semistandard Young tableau of shape lambda that are
used to describe the Schur function indexed by lambda. Descriptions of these
"Demazure" tableaux developed by the authors in earlier papers are used to
prove that the set of these tableaux is convex in Z^N if and only if pi is
R-312-avoiding if and only if the tableau set is the entire principal ideal
generated by the key of pi. These papers were inspired by results of Reiner and
Shimozono and by Postnikov and Stanley concerning coincidences between Demazure
characters and flagged Schur functions. This convexity result is used in the
next paper to deepen those results from the level of polynomials to the level
of tableau sets. The R-parabolic Catalan number is defined to be the number of
R-312-avoiding permutations. These special R-permutations are reformulated as
"R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless
R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the
publication data for the DMTCS journal (dates and volume/issue/number). This
is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v
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