The Ballot Problem with Three Candidates

Niederhausen, Heinrich

doi:10.1016/s0195-6698(83)80046-8

Cited by 25 publications

(34 citation statements)

References 3 publications

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“…One can only cite a small percentage of the relevant litterature, which dates back at least to the next-to-last century [1,20,27,33,34]. Many recent publications show that the topic is still active [4,6,12,22,24,35,36].…”

Section: Walks In the Quarter Planementioning

confidence: 99%

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Counting Walks in the Quarter Plane

Bousquet-Mélou

2002

Mathematics and Computer Science II

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Abstract. We study planar walks that start from a given point (i 0 , j 0 ), take their steps in a finite set S, and are confined in the first quadrant x ≥ 0, y ≥ 0. Their enumeration can be attacked in a systematic way: the generating function Q(x, y; t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin, this equation readsThe central question addressed in this paper is the nature of the series Q(x, y; t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself? Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovšek and the author. In both cases, we work directly from the functional equation.Keywords: enumeration, lattice walks, functional equations. Walks in the quarter planeThe enumeration of lattice walks is one of the most venerable topics in enumerative combinatorics, which has numerous applications in probability [16,30,39]. These walks take their steps in a finite subset S of Z d , and might be constrained in various ways. One can only cite a small percentage of the relevant litterature, which dates back at least to the next-to-last century [1,20,27,33,34]. Many recent publications show that the topic is still active [4,6,12,22,24,35,36].After the solution of many explicit problems, certain patterns have emerged, and a more recent trend consists in developing methods that are valid for generic sets of steps. A special attention is being paid to the nature of the generating function of the walks under consideration. For instance, the generating function for unconstrained walks on the line Z is rational, while the generating function for walks constrained to stay in the half-line N is always algebraic [3]. This result has often been described in terms of partially directed 2-dimensional walks confined in a quadrant (or generalized Dyck walks [14,21,28,29]), but is, essentially, of a 1-dimensional nature.Similar questions can be addressed for real 2-dimensional walks. Again, the generating function for unconstrained walks starting from a given point is clearly rational. Moreover, the argument used for 1-dimensional walks confined in N can be recycled to prove that the generating function for the walks that stay in the half-plane x ≥ 0 is always algebraic. What about doubly-restricted walks, that is, walks that are confined in the quadrant x ≥ 0, y ≥ 0?A rapid inspection of the most standard cases suggests that these walks might have always a D-finite generating function

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Section: Walks In the Quarter Planementioning

confidence: 99%

“…This result was first proved by Kreweras in 1965 [27, Chap. 3], and then rederived by Niederhausen [35] and Gessel [20]. It is, however, not well-understood, for two reasons: -no direct proof of (2) is known, even when i = 0.…”

Section: Walks In the Quarter Planementioning

confidence: 99%

Counting Walks in the Quarter Plane

Bousquet-Mélou

2002

Mathematics and Computer Science II

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“…It was then proved by Gessel that the associated 3-variable generating function is algebraic [26]. Since then, simpler derivations of this series have been obtained [42,13], including an automated proof [36], and a purely bijective one for walks ending at the origin [4]. See also [23,21,13], where the stationary distribution of a related Markov chain in the quarter plane is obtained, and found to be algebraic.…”

Section: Introductionmentioning

confidence: 98%

Walks with small steps in the quarter plane

Bousquet-Mélou¹,

Mishna²

2010

Algorithmic Probability and Combinatorics

159

569

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Abstract. Let S ⊂ {−1, 0, 1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study.To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases.We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function.Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.

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“…This is why Kreweras walks sometimes appear under this formulation in the literature [6]. Similarly, the meanders, that is, the walks remaining in the half-plane i + j 0, are the words w on {a, b, c} such that any prefix w of w satisfies…”

Section: Kreweras Walks and Meandersmentioning

confidence: 99%

“…The conjectured formulas were then checked using the recurrence relations between these numbers. The checking part involved several hypergeometric identities which were later simplified by Niederhausen [6]. In 1986, Gessel gave a different proof in which the guessing part was reduced [3].…”

Section: Introductionmentioning

confidence: 99%

Bijective counting of Kreweras walks and loopless triangulations

Bernardi¹

2007

Journal of Combinatorial Theory, Series A

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We consider lattice walks in the plane starting at the origin, remaining in the first quadrant i, j 0 and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations with a distinguished spanning tree. We obtain simultaneously a bijective way of counting loopless triangulations.

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The Ballot Problem with Three Candidates

Cited by 25 publications

References 3 publications

Counting Walks in the Quarter Plane

Counting Walks in the Quarter Plane

Walks with small steps in the quarter plane

Bijective counting of Kreweras walks and loopless triangulations

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