Some Explanations of Dobinski's Formula

Chen, Beifang; Yeh, Yeong‐Nan

doi:10.1002/sapm1994923191

Cited by 6 publications

(2 citation statements)

References 14 publications

(17 reference statements)

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“…Pruoj We shall prove the first of these identities, the others being similar. From the theory of exponential series [6,12,16,20,29] it follows that the derivative counts ordered trees with n + 1 vertices (as the coefficient of tfl/n!) by inversions.…”

Section: Df;(t)mentioning

confidence: 99%

Enumeration of trees by inversions

Gessel

Sagan

Yeh

1995

Journal of Graph Theory

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Mallows and Riordan "The Inversion Enumerator for Labeled Trees,"Bulletin of the American Mathematics Society, vol. 74 119681 pp. 92-94) first defined the inversion polynomial, JJ9) for trees with n vertices and found its generating function. In the present work, w e define inversion polynomials for ordered, plane, and cyclic trees, and find their values at 9 = 0, t l . Our techniques involve the use of generating functions (including Lagrange inversion), hypergeometric series, and binomial coefficient identities, induction, and bijections. We also derive asymptotic formulae for those results for which w e do not have a closed form. 0 1995 John Wiley & Sons, Inc.

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Section: Df;(t)mentioning

confidence: 99%

Enumeration of trees by inversions

Gessel

Sagan

Yeh

1995

Journal of Graph Theory

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“…Here, we make use of the mpmath arbitrary precision library for Python developed by Johansson et al (2013). This library takes advantage of Dobiǹski's Formula to approximate the Bell numbers (Dobiński, 1877;Chen and Yeh, 1994).…”

Section: Discussionmentioning

confidence: 99%

The Impact of Random Models on Clustering Similarity

Gates

Ahn

2017

Preprint

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Clustering is a central approach for unsupervised learning. After clustering is applied, the most fundamental analysis is to quantitatively compare clusterings. Such comparisons are crucial for the evaluation of clustering methods as well as other tasks such as consensus clustering. It is often argued that, in order to establish a baseline, clustering similarity should be assessed in the context of a random ensemble of clusterings. The prevailing assumption for the random clustering ensemble is the permutation model in which the number and sizes of clusters are fixed. However, this assumption does not necessarily hold in practice; for example, multiple runs of K-means clustering returns clusterings with a fixed number of clusters, while the cluster size distribution varies greatly. Here, we derive corrected variants of two clustering similarity measures (the Rand index and Mutual Information) in the context of two random clustering ensembles in which the number and sizes of clusters vary. In addition, we study the impact of one-sided comparisons in the scenario with a reference clustering. The consequences of different random models are illustrated using synthetic examples, handwriting recognition, and gene expression data. We demonstrate that the choice of random model can have a drastic impact on the ranking of similar clustering pairs, and the evaluation of a clustering method with respect to a random baseline; thus, the choice of random clustering model should be carefully justified.

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Computation of certain infinite series of the form for arbitrary real-valued k

Gorder

2009

Applied Mathematics and Computation

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Some Explanations of Dobinski's Formula

Abstract: The geometric, algebraic, and combinatorial explanations of Dobinski's formula are presented by mixed volumes of compact convex sets, Mobius inversion, difference operator, and species. The employed method may be useful in proving some other combinatorial identities.

Cited by 6 publications

References 14 publications

Enumeration of trees by inversions

Enumeration of trees by inversions

The Impact of Random Models on Clustering Similarity

Computation of certain infinite series of the form for arbitrary real-valued k

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