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Log-convexity and the cycle index polynomials with relation to compound Poisson distributions
Muneya Matsui · 2016-09-22 · via math.ST updates on arXiv.org

We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and binomial convolutions of sequences which satisfy the exponential formula. The operations generate interesting examples which are not included in well known laws about log-concavity/convexity.

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