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Abstract: Using methods from Analytic Combinatorics, we study the families of perfect matchings, partitions, chord diagrams, and hyperchord diagrams on a disk with a prescribed number of crossings. For each family, we express the generating function of the configurations with exactly k crossings as a rational function of the generating function of crossing-free configurations. Using these expressions, we study the singular behavior of these generating functions and derive asymptotic results on the counting sequences of the configurations with precisely k crossings. Limiting distributions and random generators are also studied. [Copyright &y& Elsevier]

Abstract: Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi–Stirling numbers and is a Pólya frequency sequence if and only if and study the z-total positivity properties of these numbers. Moreover, the polynomial sequences are proved to be strongly -log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapotonʼs generalized Ramanujan polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising from the Lambert W function, we obtain a neat proof of the unimodality of the latter sequence, which was proved previously by Kalugin and Jeffrey. [Copyright &y& Elsevier]

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 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence is log-concave. Furthermore, we prove that if a sequence is ratio log-concave, then the sequence is strictly log-concave subject to an 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 , we confirm a conjecture of Sun on the log-concavity of the sequence . [Copyright &y& Elsevier]

Abstract: The Jacobi–Stirling numbers of the first and second kinds were first introduced in Everitt et al. (2007) and they are a generalization of the Legendre–Stirling numbers. Quite remarkably, they share many similar properties with the classical Stirling numbers. In this paper we study total positivity properties of these numbers. In particular, we prove that the matrix whose entries are the Jacobi–Stirling numbers is totally positive and that each row and each column is a Pólya frequency sequence, except for the columns with (unsigned) numbers of the first kind. [Copyright &y& Elsevier]

Abstract: We describe various aspects of the Al-Salam–Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells. [Copyright &y& Elsevier]

Abstract: A well-known theorem of Knuth establishes a bijection between permutations in with no decreasing subsequence of length three and rectangular standard Young tableaux of shape . We prove an asymptotic version of this result: for any fixed integer , the number of permutations in with no decreasing subsequence of length is asymptotically equal, as , to the number of standard Young tableaux on the rectangle . This yields a new proof of Regevʼs theorem on the asymptotic number of permutations without long decreasing subsequences, and consequently an alternative, elementary evaluation of Mehtaʼs integral at . [Copyright &y& Elsevier]

Abstract: L. Shapiro found an elegant formula for the self-convolution of the even subscripted terms in the Catalan sequence. This paper provides a natural q-analog of Shapiroʼs formula together with three proofs, one of which is purely combinatorial. [Copyright &y& Elsevier]

Abstract: We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence x is generated by two independent substitutions, at least one being of exponential growth, then the factors of x appearing infinitely often in x appear with bounded gaps. As an application, we derive an analogue of Cobham''s theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential. [Copyright &y& Elsevier]