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We study the Dyson rank function N (r, 3; n), the number of partitions of n with rank ≡ r (mod 3). We investigate the convexity of these functions. We extend N (r, 3; n) multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size n.

We study the tradeoff between social welfare maximization and fairness in the context of ad auctions. We study an ad auction setting where users arrive in an online fashion, k advertisers submit bids for each user, and the auction assigns a distribution over ads to the user. Following the works of Dwork and Ilvento [10] and Chawla et al. [12], our goal is to design a truthful auction that satisfies multiple-task fairness in its outcomes: informally speaking, users that are similar to each other should obtain similar allocations of ads.We develop a new class of allocation algorithms that we call inverse-proportional allocation. These allocation algorithms are truthful, online, and do not explicitly need to know the fairness constraint over the users. In terms of fairness, they guarantee fair outcomes as long as every advertiser's bids are non-discriminatory across users. In terms of social welfare, inverse-proportional allocation achieves a constant factor approximation in social welfare against the optimal (unfair) allocation, independent of the number of advertisers in the system. In this respect, these allocation algorithms greatly surpass the guarantees achieved in previous work; in fact, they achieve the optimal tradeoffs between fairness and social welfare in some contexts. We also extend our results to broader notions of fairness that we call subset fairness.

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