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Arxiv link: https://arxiv.org/abs/1807.06874
This a nice paper and I think you manage to keep your article easily readable. Specially the figures, such as figure 3, helped me follow your explanations. I haven't finished reading the paper but I have minor remarks/typo: - Page 9 in the definition of multigraph, should e(v, V) be transformed to e({v},V) in order to be completely rigorous? - I don't know what “maximum-entropy sampling” is but why quoting it ? - I don't get the use of Reversible Decompression, apart from being an extreme case of external information. I don't think this case deserve to be explained in detailed, as it does not have any impact on the understanding of the paper. - I find the notation for definition 10 a bit confusing. A Cartesian MultiEdge Partition is made of Cartesian products of two vertex subsets. The notation for each cartesian product uses subscript on vertex subset: $(V_i \times V'_i)$ for the $i$th product. This lead me to think that $(V_i \times V'_i)$ was a possible Cartesian product. This also lead me to think that Cartesian MultiEdge Partition are made of Cartesian product of vertex partition: $B(V \times V) = \{(V_i \times V_j)\}\ \forall V_i \in P_1, V_j \in P_2 \wedge P_1 \in B(V), P_2 \in B(V)$. I think all Cartesian product of vertex partition are Cartesian MultiEdge partition. Is there any Cartesian MultiEdge Partition that is not a Cartesian product of vertex partition? - In the conplete case after definition 11, I don't know the notation $\omega$ for the number of feasible partitions. - I think you missed a $\lambda$ in the lagrange function page 21. Should it be $q_{\lambda}(VVT)= |VVT| + \lambda Loss(VVT)$
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One of my favorite papers. Related to [Axioms for centrality](https://papers-gamma.link/paper/54)
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Un jour, j'étais très heureux d'avoir été invité par Brigitte Vallée au séminaire [ALGO](https://clementj01.users.greyc.fr/semalgo/) afin de présenter nos travaux, intitulées [Patterns in treeshelves](https://kirgizov.link/publications/boxed_prod_DM.pdf). Le soir du 28 Février 2017, après le séminaire, dans le train de Caen à Dijon via Paris je pensais à la façon dont nous pouvons à partir de chemins de Dyck construire les graphes avec une certaine structure de clusters-communautés. A mi-chemin vers cet objectif, et à mi-chemin vers Dijon, j'ai trouvé une sous-classe très intéressante de chemins de Dyck comptés par les nombres de Motzkin ! Ce voyage et la discussions ultérieures avec Jean-Luc Baril et Armen Petrossian ont donné naissance à nouvel article.
Feel free to discuss this paper that was recently accepted to publication in [Discrete Mathematics Jounral](https://en.wikipedia.org/wiki/Discrete_Mathematics_%28journal%29).
The paper is [published](https://www.sciencedirect.com/science/article/pii/S0012365X18300670)!
Il s’agit d’un très jolie classe de chemins de Dyck énumérés par les nombres de Motzkin. 👍🏼👍🏼👍🏼
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