45 research outputs found
Dyck paths with coloured ascents
We introduce a notion of Dyck paths with coloured ascents. For several ways
of colouring, we establish bijections between sets of such paths and other
combinatorial structures, such as non-crossing trees, dissections of a convex
polygon, etc. In some cases enumeration gives new expression for sequences
enumerating these structures.Comment: 14 pages, 11 figure
Old and young leaves on plane trees
A leaf of a plane tree is called an old leaf if it is the leftmost child of
its parent, and it is called a young leaf otherwise. In this paper we enumerate
plane trees with a given number of old leaves and young leaves. The formula is
obtained combinatorially by presenting two bijections between plane trees and
2-Motzkin paths which map young leaves to red horizontal steps, and old leaves
to up steps plus one. We derive some implications to the enumeration of
restricted permutations with respect to certain statistics such as pairs of
consecutive deficiencies, double descents, and ascending runs. Finally, our
main bijection is applied to obtain refinements of two identities of Coker,
involving refined Narayana numbers and the Catalan numbers.Comment: 11 pages, 7 figure
Counting outerplanar maps
A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our
results, we obtain an e cient scheme for encoding simple outerplanar maps.Peer ReviewedPostprint (published version
Flip-sort and combinatorial aspects of pop-stack sorting
Flip-sort is a natural sorting procedure which raises fascinating
combinatorial questions. It finds its roots in the seminal work of Knuth on
stack-based sorting algorithms and leads to many links with permutation
patterns. We present several structural, enumerative, and algorithmic results
on permutations that need few (resp. many) iterations of this procedure to be
sorted. In particular, we give the shape of the permutations after one
iteration, and characterize several families of permutations related to the
best and worst cases of flip-sort. En passant, we also give some links between
pop-stack sorting, automata, and lattice paths, and introduce several tactics
of bijective proofs which have their own interest.Comment: This v3 just updates the journal reference, according to the
publisher wis
Intervals in the greedy Tamari posets
We consider a greedy version of the -Tamari order defined on -Dyck
paths, recently introduced by Dermenjian. Inspired by intriguing connections
between intervals in the ordinary 1-Tamari order and planar triangulations, and
more generally by the existence of simple formulas counting intervals in the
ordinary -Tamari orders, we investigate the number of intervals in the
greedy order on -Dyck paths of fixed size. We find again a simple formula,
which also counts certain planar maps (of prescribed size) called
-constellations.
For instance, when the number of intervals in the greedy order on
1-Dyck paths of length is proved to be , which is also the number of bipartite maps with edges.
Our approach is recursive, and uses a ``catalytic'' parameter, namely the
length of the final descent of the upper path of the interval. The resulting
bivariate generating function is algebraic for all . We show that the same
approach can be used to count intervals in the ordinary -Tamari lattices as
well. We thus recover the earlier result of the first author, Fusy and
Pr\'eville-Ratelle, who were using a different catalytic parameter.Comment: 23 page