Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson-Schensted-like insertion/deletion algorithms, we use the growth diagram construction of Fomin to obtain our results. We extend the results by Chen et al., which, in the language of fillings, are results about $0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very likely, these results are part of a bigger picture which also includes recent results of Jonsson on $0$-$1$-fillings of stack polyominoes, and of results of Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the enumeration of permutations and involutions with restricted patterns. In particular, we show that our growth diagram bijections do in fact provide alternative proofs of the results by Backelin, West and Xin and by Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob Jonsson, the final section is now much more profound and has additional result

Continued fractions for permutation statistics

We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle diagrams, we find simple translations of some statistics on permutations (and subsets of permutations) into statistics on colored Motzkin paths, which are amenable to the use of continued fractions. We obtain new enumeration formulas for subsets of permutations with respect to fixed points, excedances, double excedances, cycles, and inversions. In particular, we prove that cyclic permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC

Filtering With the Crowd: CrowdScreen Revisited

Filtering a set of items, based on a set of properties that can be verified by humans, is a common application of CrowdSourcing. When the workers are error-prone, each item is presented to multiple users, to limit the probability of misclassification. Since the Crowd is a relatively expensive resource, minimizing the number of questions per item may naturally result in big savings. Several algorithms to address this minimization problem have been presented in the CrowdScreen framework by Parameswaran et al. However, those algorithms do not scale well and therefore cannot be used in scenarios where high accuracy is required in spite of high user error rates. The goal of this paper is thus to devise algorithms that can cope with such situations. To achieve this, we provide new theoretical insights to the problem, then use them to develop a new efficient algorithm. We also propose novel optimizations for the algorithms of CrowdScreen that improve their scalability. We complement our theoretical study by an experimental evaluation of the algorithms on a large set of synthetic parameters as well as real-life crowdsourcing scenarios, demonstrating the advantages of our solution