82 research outputs found
Generating functions for generating trees
Certain families of combinatorial objects admit recursive descriptions in
terms of generating trees: each node of the tree corresponds to an object, and
the branch leading to the node encodes the choices made in the construction of
the object. Generating trees lead to a fast computation of enumeration
sequences (sometimes, to explicit formulae as well) and provide efficient
random generation algorithms. We investigate the links between the structural
properties of the rewriting rules defining such trees and the rationality,
algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the
article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and
published in its vol. 246(1-3), March 2002, pp. 29-5
Solving multivariate functional equations
This paper presents a new method to solve functional equations of
multivariate generating functions, such as
giving a
formula for in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for
greater clarity. Updated to final version, to appear in Discrete Mathematic
A closed formula for the number of convex permutominoes
In this paper we determine a closed formula for the number of convex
permutominoes of size n. We reach this goal by providing a recursive generation
of all convex permutominoes of size n+1 from the objects of size n, according
to the ECO method, and then translating this construction into a system of
functional equations satisfied by the generating function of convex
permutominoes. As a consequence we easily obtain also the enumeration of some
classes of convex polyominoes, including stack and directed convex
permutominoes
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
A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape
Tableau sequences of bounded height have been central to the analysis of
k-noncrossing set partitions and matchings. We show here that familes of
sequences that end with a row shape are particularly compelling and lead to
some interesting connections. First, we prove that hesitating tableaux of
height at most two ending with a row shape are counted by Baxter numbers. This
permits us to define three new Baxter classes which, remarkably, do not
obviously possess the antipodal symmetry of other known Baxter classes. We then
conjecture that oscillating tableau of height bounded by k ending in a row are
in bijection with Young tableaux of bounded height 2k. We prove this conjecture
for k at most eight by a generating function analysis. Many of our proofs are
analytic in nature, so there are intriguing combinatorial bijections to be
found.Comment: 10 pages, extended abstrac
Words in Linear Groups, Random Walks, Automata and P-Recursiveness
Fix a finite set . Denote by the number
of products of matrices in of length that are equal to 1. We show that
the sequence is not always P-recursive. This answers a question of
Kontsevich.Comment: 10 pages, 1 figur
Tournament Sequences and Meeussen Sequences
A "tournament sequence" is an increasing sequence of positive integers
(t_1,t_2,...) such that t_1=1 and t_{i+1} <= 2 t_i. A "Meeussen sequence" is an
increasing sequence of positive integers (m_1,m_2,...) such that m_1=1, every
nonnegative integer is the sum of a subset of the {m_i}, and each integer m_i-1
is the sum of a unique such subset.
We show that these two properties are isomorphic. That is, we present a
bijection between tournament and Meeussen sequences which respects the natural
tree structure on each set. We also present an efficient technique for counting
the number of tournament sequences of length n, and discuss the asymptotic
growth of this number. The counting technique we introduce is suitable for
application to other well-behaved counting problems of the same sort where a
closed form or generating function cannot be found.Comment: 16 pages, 1 figure. Minor changes only; final version as published in
EJ
Restricted ascent sequences and Catalan numbers
Ascent sequences are those consisting of non-negative integers in which the
size of each letter is restricted by the number of ascents preceding it and
have been shown to be equinumerous with the (2+2)-free posets of the same size.
Furthermore, connections to a variety of other combinatorial structures,
including set partitions, permutations, and certain integer matrices, have been
made. In this paper, we identify all members of the (4,4)-Wilf equivalence
class for ascent sequences corresponding to the Catalan number
C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance
of a single pattern and provides apparently new combinatorial interpretations
for C_n. In several cases, the subset of the class consisting of those members
having exactly m ascents is given by the Narayana number
N_{n,m+1}=\frac{1}{n}\binom{n}{m+1}\binom{n}{m}.Comment: 12 page
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