Enumerations of plane trees with multiple edges and Raney lattice paths

Dziemiańczuk, Maciej

doi:10.1016/j.disc.2014.07.024

Cited by 9 publications

(14 citation statements)

References 11 publications

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“…the paragraph following [3, Theorem 3.13] with k = r, φ k = d k , Φ(y) = (1 + y) d , τ = 1/(d − 1)), which tends to 1/(r!e) as d → ∞. This generalises the observation made in [4] in the case r = 0 that the asymptotic average proportion of leaves tends to 1/e as d → ∞.…”

Section: A Bijection Between D-ary Multi-edge Trees and Ordinary D-arsupporting

confidence: 81%

“…Dziemiańczuk [4] has introduced a tree model based on plane (=planar) trees [10, p. 31], which are enumerated by Catalan numbers. Instead of connecting two vertices by one edge, in his multi-edge model, two vertices can be connected by several edges.…”

Section: Introductionmentioning

confidence: 99%

“…If one counts trees by vertices, one must somehow restrict the number of edges in order to avoid an infinity of objects with the same number of vertices. In [4], the chosen restriction is that each vertex has out-degree at most d, i.e., there are at most d edges going out from any vertex. However, if one counts trees with a given number of edges, the restriction with the parameter d is no longer necessary.…”

Section: Introductionmentioning

confidence: 99%

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The height of multiple edge plane trees

2015

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Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically

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Section: A Bijection Between D-ary Multi-edge Trees and Ordinary D-arsupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The height of multiple edge plane trees

2015

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“…for any fixed N ≥ 0 and K ≥ 1. It is worth pointing out that B-paths, C-paths, D-paths, and E-path are generalized Lukasiewicz paths, Raney paths recently considered by the author in [8], generalized Dyck paths, and Delannoy paths, respectively.…”

Section: Introductionmentioning

confidence: 99%

On directed lattice paths with vertical steps

Dziemiańczuk

2016

Discrete Mathematics

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The paper is devoted to the study of lattice paths that consist of vertical steps (0, −1) and non-vertical steps (1, k) for some k ∈ Z. Two special families of primary and free lattice paths with vertical steps are considered. It is shown that for any family of primary paths there are equinumerous families of proper weighted lattice paths that consist of only non-vertical steps. The relation between primary and free paths is established and some combinatorial and statistical properties are obtained. It is shown that the expected number of vertical steps in a primary path running from (0, 0) to (n, −1) is equal to the number of free paths running from (0, 0) to (n, 0). Enumerative results with generating functions are given. Finally, a few examples of families of paths with vertical steps are presented and related to Lukasiewicz, Motzkin, Dyck and Delannoy paths.

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“…Interestingly Penson and Zyczkowski were the first who use the term Raney numbers [ [3] and a bijection exists between Raney path and plan multitree [4]. These numbers do not form novel sequences, as the numbers were introduced earlier as a generalization of the binomial series [5].…”

Section: Introductionmentioning

confidence: 99%