57,030 research outputs found
Avoiding three consecutive blocks of the same size and same sum
We show that there exists an infinite word over the alphabet {0,1,3,4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from1994
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
Van der Waerden's Theorem and Avoidability in Words
Pirillo and Varricchio, and independently, Halbeisen and Hungerbuhler
considered the following problem, open since 1994: Does there exist an infinite
word w over a finite subset of Z such that w contains no two consecutive blocks
of the same length and sum? We consider some variations on this problem in the
light of van der Waerden's theorem on arithmetic progressions.Comment: Co-author added; new result
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
Clusters, generating functions and asymptotics for consecutive patterns in permutations
We use the cluster method to enumerate permutations avoiding consecutive
patterns. We reprove and generalize in a unified way several known results and
obtain new ones, including some patterns of length 4 and 5, as well as some
infinite families of patterns of a given shape. By enumerating linear
extensions of certain posets, we find a differential equation satisfied by the
inverse of the exponential generating function counting occurrences of the
pattern. We prove that for a large class of patterns, this inverse is always an
entire function. We also complete the classification of consecutive patterns of
length up to 6 into equivalence classes, proving a conjecture of Nakamura.
Finally, we show that the monotone pattern asymptotically dominates (in the
sense that it is easiest to avoid) all non-overlapping patterns of the same
length, thus proving a conjecture of Elizalde and Noy for a positive fraction
of all patterns
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