9,212 research outputs found
Snow Leopard Permutations and Their Even and Odd Threads
Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard
permutations, which are the anti-Baxter permutations that are compatible with
the doubly alternating Baxter permutations. Among other things, they showed
that these permutations preserve parity, and that the number of snow leopard
permutations of length is the Catalan number . In this paper we
investigate the permutations that the snow leopard permutations induce on their
even and odd entries; we call these the even threads and the odd threads,
respectively. We give recursive bijections between these permutations and
certain families of Catalan paths. We characterize the odd (resp. even) threads
which form the other half of a snow leopard permutation whose even (resp. odd)
thread is layered in terms of pattern avoidance, and we give a constructive
bijection between the set of permutations of length which are both even
threads and odd threads and the set of peakless Motzkin paths of length .Comment: 25 pages, 6 figures. Version 3 is modified to use standard Discrete
Mathematics and Theoretical Computer Science but is otherwise unchange
Baxter permutations rise again
AbstractBaxter permutations, so named by Boyce, were introduced by Baxter in his study of the fixed points of continuous functions which commute under composition. Recently Chung, Graham, Hoggatt, and Kleiman obtained a sum formula for the number of Baxter permutations of 2n − 1 objects, but admit to having no interpretation of the individual terms of this sum. We show that in fact the kth term of this sum counts the number of (reduced) Baxter permutations that have exactly k − 1 rises
Bijections for Baxter Families and Related Objects
The Baxter number can be written as . These
numbers have first appeared in the enumeration of so-called Baxter
permutations; is the number of Baxter permutations of size , and
is the number of Baxter permutations with descents and
rises. With a series of bijections we identify several families of
combinatorial objects counted by the numbers . Apart from Baxter
permutations, these include plane bipolar orientations with vertices and
faces, 2-orientations of planar quadrangulations with white and
black vertices, certain pairs of binary trees with left and
right leaves, and a family of triples of non-intersecting lattice paths. This
last family allows us to determine the value of as an
application of the lemma of Gessel and Viennot. The approach also allows us to
count certain other subfamilies, e.g., alternating Baxter permutations, objects
with symmetries and, via a bijection with a class of plan bipolar orientations
also Schnyder woods of triangulations, which are known to be in bijection with
3-orientations.Comment: 31 pages, 22 figures, submitted to JCT
Baxter permutations and plane bipolar orientations
We present a simple bijection between Baxter permutations of size and
plane bipolar orientations with n edges. This bijection translates several
classical parameters of permutations (number of ascents, right-to-left maxima,
left-to-right minima...) into natural parameters of plane bipolar orientations
(number of vertices, degree of the sink, degree of the source...), and has
remarkable symmetry properties. By specializing it to Baxter permutations
avoiding the pattern 2413, we obtain a bijection with non-separable planar
maps. A further specialization yields a bijection between permutations avoiding
2413 and 3142 and series-parallel maps.Comment: 22 page
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
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