5 research outputs found
Lattice paths of slope 2/5
We analyze some enumerative and asymptotic properties of Dyck paths under a
line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet
lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in
June 2014.Our approach relies on the work of Banderier and Flajolet for
asymptotics and enumeration of directed lattice paths. A key ingredient in the
proof is the generalization of an old trick of Knuth himself (for enumerating
permutations sortable by a stack),promoted by Flajolet and others as the
"kernel method". All the corresponding generating functions are algebraic,and
they offer some new combinatorial identities, which can be also tackled in the
A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar
results for other slopes than 2/5, an interesting case being e.g. Dyck paths
below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and
Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO
Pattern avoidance in forests of binary shrubs
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line y = lx, for some l in Q+, one of these being the celebrated Duchon’s club paths with l = 2/3. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate