Abstract. We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {a n } n≥0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {a n+1 /a n } n≥0 is log-concave. Furthermore, we prove that if a sequence {a n } n≥k is ratio log-concave, then the sequence { n √ a n } n≥k is strictly logconcave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers D n , we confirm a conjecture of Sun on the log-concavity of the sequence { n √ D n } n≥1 .

In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic sor, called the sorting index. Petersen proved that the pairs of statistics (sor, cyc) and (inv, rl-min) have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, and we show that the bijection of Foata and Han serves the purpose of mapping (inv, rl-min) to (sor, cyc). We also give a type B analogue of the Foata-Han bijection, and we derive the equidistribution of (inv B , Lmap B , Rmil B ) and (sor B , Lmap B , Cyc B ) over signed permutations. So we get a combinatorial interpretation of Petersen's equidistribution of (inv B , nmin B ) and (sor B , l ′ B ). Moreover, we show that the six pairs of set-valued statistics (Cyc B , Rmil B ), (Cyc B , Lmap B ), (Rmil B , Lmap B ), (Lmap B , Rmil B ), (Lmap B , Cyc B ) and (Rmil B , Cyc B ) are equidistributed over signed permutations.For Coxeter groups of type D, Petersen showed that the two statistics inv D and sor D are equidistributed. We introduce two statistics nmin D andl ′ D for elements of D n and we prove that the two pairs of statistics (inv D , nmin D ) and (sor D ,l ′ D ) are equidistributed.

In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζ µ (x) to study the log-behavior of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B 2n |/(2n)!} n≥1 is log-convex, where B n is the n-th Bernoulli number. We introduce the function θ(is the gamma function, and we show that log θ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence { n |B 2n |} n≥1 is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers a n (µ) = 2 2n+1 (n+1)!(µ+1) n ζ µ (2n) and conjectured that the sequence {a n (µ)} n≥1 is log-convex for µ = 0 and µ = 1. By proving that ζ µ (x) is log-convex for x > 1 and µ > −1, we show that the sequence {a n (µ)} n≥1 is log-convex for any µ > −1. We introduce another function θ µ (x) involving ζ µ (x) and the gamma function Γ(x) and we show that log θ µ (x) is strictly increasing for x > 8e(µ + 2) 2 . This implies that n a n (µ) < n+1 a n+1 (µ) for n > 4e(µ+2) 2 . Based on Dobinski's formula, we prove that n √ B n < n+1 √ B n+1 for n ≥ 1, where B n is the n-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of { n √ B n } n≥1 and Hölder's inequality in probability theory.

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