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A bijection of plane increasing trees with relaxed binary trees of right height at most one

THEORETICAL COMPUTER SCIENCE 2019年 755卷 1-12页

作者： Wallner, Michael Acad Sinica Inst Stat Sci 128 Acad Rd Taipei 11529 Taiwan

Plane increasing trees are rooted labeled trees embedded into the plane such that the sequence of labels is increasing on any branch starting at the root. Relaxed binary trees are a subclass of unlabeled directed acyclic graphs. We construct a bijection between these two combinatorial objects and study the therefrom arising connections of certain parameters. Furthermore, we show central limit theorems for two statistics on leaves. We end the study by considering more than 20 subclasses and their bijective counterparts. Many of these subclasses are enumerated by known counting sequences, and thus enrich their combinatorial interpretation. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Bijection Directed acyclic graphs Increasing trees Fibonacci numbers Analytic combinatorics Limit laws random generation

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Some bijective results about the area of Schroder paths

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THEORETICAL COMPUTER SCIENCE 2003年第2期307卷 327-335页

作者： Ferrari, L Grazzini, E Pergola, E Rinaldi, S Univ Florence Dipartimento Matemat U Dini I-50134 Florence Italy Univ Florence Dipartimento Sistemi & Informat I-50134 Florence Italy Dipartimento Matemat R Magari I-53100 Siena Italy

In the existing literature there are many combinatorial interpretations for the sequence (a(i))(i greater than or equal to 1) = 1, 3, 7, 17, 41,... (M2665 in The Encyclopedia of Integer Sequences, Academic Press, New York, 1995), and at least one for the subsequence of its odd-indexed terms, i.e. 1, 7, 41, 239,..., using the area under elevated Schroder paths. We provide a combinatorial interpretation for the subsequence given by the remaining terms 3, 17, 99,..., also in this case by using the area under Schroder paths. (C) 2003 Elsevier B.V. All rights reserved.

关键词： schroder paths self-avoiding paths

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Two bijective proofs for the arborescent form of the Good-Lagrange formula and some applications to colored rooted trees and cacti

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THEORETICAL COMPUTER SCIENCE 2003年第2期307卷 277-302页

作者： Bousquet, M Chauve, C Labelle, G Leroux, P Univ Quebec LaCIM Montreal PQ H3C 3P8 Canada

Goulden and Kulkarni (J. Combin. Theory Ser. A 80 (2) (1997) 295) give a bijective proof of an arborescent form of the Good-Lagrange multivariable inversion formula, This formula was first stated explicitly by Bender and Richmond (Electron. J. Combin. 5 (1) (1998) 4pp) but is implicit in Goulden and Kulkarni (1997). In this paper, we propose two new simple bijective proofs of this formula and we illustrate the interest of these proofs by applying them to the enumeration and random generation of colored rooted trees and rooted m-ary cacti. (C) 2003 Elsevier B.V. All rights reserved.

关键词： enumeration random generation multisort species multivariable power series Good-Lagrange formula bijections trees-like structures

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A loopless algorithm for generating the permutations of a multiset

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THEORETICAL COMPUTER SCIENCE 2003年第2期307卷 415-431页

作者： Vajnovszki, V Univ Bourgogne CNRS FRE 2309 LE2I F-21078 Dijon France

Many combinatorial structures can be constructed from simpler components. For example, a permutation can be constructed from cycles, or a Motzkin word from a Dyck word and a combination. In this paper we present a constructor for combinatorial structures, called shuffle oil trajectories (defined previously in a non-combinatorial context), and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects. (C) 2003 Elsevier B.V. All rights reserved.

关键词： gray codes loopless generating algorithms multiset permutations shuffle combinatorial objects

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A loopless algorithm for generating the permutations of a multiset

A loopless algorithm for generating the permutations of a mu...

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conference on random generation of combinatorial objects and bijective combinatorics

作者： Vajnovszki, V Univ Bourgogne CNRS FRE 2309 LE2I F-21078 Dijon France

关键词： gray codes loopless generating algorithms multiset permutations shuffle combinatorial objects

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Watermelon uniform random generation with applications

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THEORETICAL COMPUTER SCIENCE 2003年第2期307卷 241-256页

作者： Bonichon, N Mosbah, M Univ Bordeaux 1 LaBRI F-33405 Talence France

Watermelons are particular configurations of vicious walkers. In these configurations, each path starts and ends at the same ordinate. We present a simple uniform random generation algorithm of watermelons based on enumeration formulas of star configurations (with or without a wall). The performance of this algorithm is better than earlier ones in the case of watermelons with few walkers. Using appropriate bijections, these algorithms can also generate underdiagonal paths, realizers (or Schnyder Trees), twin parallelogram polyominoes according to their perimeter and width, Baxter permutations according to the number of rises, etc. Moreover, we present some experimental results on the height of watermelons and realizers. (C) 2003 Elsevier B.V. All rights reserved.

关键词： vicious walkers watermelon uniform random realizer Baxter permutation

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Two bijective proofs for the arborescent form of the Good-Lagrange formula and some applications to colored rooted trees and cacti

Two bijective proofs for the arborescent form of the Good-La...

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conference on random generation of combinatorial objects and bijective combinatorics

作者： Bousquet, M Chauve, C Labelle, G Leroux, P Univ Quebec LaCIM Montreal PQ H3C 3P8 Canada

关键词： enumeration random generation multisort species multivariable power series Good-Lagrange formula bijections trees-like structures

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Some bijective results about the area of Schroder paths

Some bijective results about the area of Schroder paths

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conference on random generation of combinatorial objects and bijective combinatorics

关键词： schroder paths self-avoiding paths

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Watermelon uniform random generation with applications

Watermelon uniform random generation with applications

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conference on random generation of combinatorial objects and bijective combinatorics

作者： Bonichon, N Mosbah, M Univ Bordeaux 1 LaBRI F-33405 Talence France

关键词： vicious walkers watermelon uniform random realizer Baxter permutation

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Walks confined in a quadrant are not always D-finite

Walks confined in a quadrant are not always D-finite

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conference on random generation of combinatorial objects and bijective combinatorics

作者： Bousquet-Mélou, M Petkovsek, M Univ Bordeaux 1 LabRI CNRS F-33405 Talence France Univ Ljubljana Dept Math SI-1000 Ljubljana Slovenia

We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z(2), and always stay in the quadrant x greater than or equal to 0, y greater than or equal to 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (I I), take their steps in {(2, -1),(-1, 2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis. (C) 2003 Elsevier B.V. All rights reserved.

关键词： combinatorial mathematics

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