Combinatorial parameters in bargraphs

Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold

doi:10.2989/16073606.2015.1121932

Cited by 33 publications

(20 citation statements)

References 11 publications

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“…This was described initially by Prellberg and Brak in [16] and more recently in [2], where it is called the wasp-waist decomposition. The present authors have also discussed it in [1].…”

Section: Introductionmentioning

confidence: 73%

“…It follows from the wasp-waist decomposition that the generating function B(x, y) which counts all bargraphs is In [1,2] the authors found an asymptotic expression for B(z, z), where z marks the semiperimeter of the bargraphs. This is known as the generating function for the isotropic case.…”

Section: Introductionmentioning

confidence: 99%

“…Wasp-waist decomposition of bargraphsHere x counts the number of horizontal steps and y counts the number of up steps (see Theorem 1 in[1]) or[2,7]). The series expansion, B(x, y) beginsx(y + y 2 + y 3 + y 4 ) + x 2 (y + 3y 2 + 5y 3 + 7y 4 ) + x 3 (y + 6y 2 + 16y 3 + 31y 4 ) + x 4 (y + 10y 2 + 40y 3 + 105y 4 + 219y 4 ).The bold coefficient of x 4 y 2 is illustrated below with the full set of 10 bargraphs with 4 horizontal steps and 2 vertical up steps.…”

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confidence: 99%

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Levels in bargraphs

2015

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Bargraphs are lattice paths in N 2 0 , which start at the origin and terminate immediately upon return to the x-axis. The allowed steps are the up step (0, 1), the down step (0, −1) and the horizontal step (1, 0). The first step is an up step and the horizontal steps must all lie above the x-axis. An up step cannot follow a down step and vice versa. In this paper we consider levels, which are maximal sequences of two or more adjacent horizontal steps. We find the generating functions that count the total number of levels, the leftmost x-coordinate and the height of the first level and obtain the generating function for the mean of these parameters. Finally, we obtain the asymptotics of these means as the length of the path tends to infinity.

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“…This was described initially by Prellberg and Brak in [16] and more recently in [2], where it is called the wasp-waist decomposition. The present authors have also discussed it in [1].…”

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Levels in bargraphs

2015

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“…As a special class of convex polyominoes, they have been studied and enumerated by Bousquet-Mélou and Rechnitzer [6], Prellberg and Brak [10], and Feretić [8]. The distribution of several bargraph statistics has been given in a series of papers by Blecher, Brennan, Knopfmacher and Prodinger [1,2,3,4,5]. All of the above papers rely on a decomposition of bargraphs that is used to obtain equations satisfied by the corresponding generating functions.…”

Section: Introductionmentioning

confidence: 99%

A bijection between bargraphs and Dyck paths

Deutsch¹,

Elizalde

2018

Discrete Applied Mathematics

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Bargraphs are a special class of convex polyominoes. They can be identified with lattice paths with unit steps north, east, and south that start at the origin, end on the x-axis, and stay strictly above the x-axis everywhere except at the endpoints. Bargraphs, which are used to represent histograms and to model polymers in statistical physics, have been enumerated in the literature by semiperimeter and by several other statistics, using different methods such as the wasp-waist decomposition of Bousquet-Mélou and Rechnitzer, and a bijection with certain Motzkin paths.In this paper we describe an unusual bijection between bargraphs and Dyck paths, and study how some statistics are mapped by the bijection. As a consequence, we obtain a new interpretation of Catalan numbers, as counting bargraphs where the semiperimeter minus the number of peaks is fixed.

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“…These papers use the so-called wasp-waist decomposition or some variation of it to find a generating function with two variables x and y keeping track of the number of H and U steps, respectively. The same decomposition has later been exploited by Blecher, Brennan and Knopfmacher to obtain refined enumerations with respect to statistics such as peaks [2], levels [1], walls [4], descents and area [3]. Bargraphs are used to represent histograms and they also have connections to statistical physics, where they are used to model polymers.…”

Section: Introductionmentioning

confidence: 99%

Statistics on bargraphs viewed as cornerless Motzkin paths

Deutsch¹,

Elizalde

2017

Discrete Applied Mathematics

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A bargraph is a self-avoiding lattice path with steps U = (0, 1), H = (1, 0) and D = (0, −1) that starts at the origin and ends on the x-axis, and stays strictly above the x-axis everywhere except at the endpoints. Bargraphs have been studied as a special class of convex polyominoes, and enumerated using the so-called wasp-waist decomposition of Bousquet-Mélou and Rechnitzer. In this paper we note that there is a trivial bijection between bargraphs and Motzkin paths without peaks or valleys. This allows us to use the recursive structure of Motzkin paths to enumerate bargraphs with respect to several statistics, finding simpler derivations of known results and obtaining many new ones. We also count symmetric bargraphs and alternating bargraphs. In some cases we construct statistic-preserving bijections between different combinatorial objects, proving some identities that we encounter along the way.

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Combinatorial parameters in bargraphs

Cited by 33 publications

References 11 publications

Levels in bargraphs

Levels in bargraphs

A bijection between bargraphs and Dyck paths

Statistics on bargraphs viewed as cornerless Motzkin paths

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