Enumerative aspects of secondary structures

Seven (Lattice) Paths to Log-Convexity

“…Although we have demonstrated them on rather short recurrences, there is nothing that restricts the length of a recurrence (at least in principle). (For example, the interlacing method has been successfully applied to a certain sequence given by a recurrence of length 10 [23].) Also, there is no reason to require the coefficient functions to be rational.…”

“…However, neither of those proofs was suitable for proving the log-convexity of closely related sequences enumerating secondary structures of single-stranded nucleic acids (see, e.g., [23]). This prompted a search for other methods, and resulted in developing three similar methods, which we describe here as the calculus method, the finite difference method and the interlacing method.…”

“…The literature on log-concavity and related issues, such as total positivity and unimodality, is vast; we refer the interested reader to [45] and [15] for comprehensive surveys, and to [11,14,28,39], and [23] for illustrations of various techniques of establishing logconcavity. On the other hand, log-convexity has so far attracted much less attention.…”

“…This, in turn, means that the sequence x n is increasing, and that the sequence W n is log-convex. The interlacing method has been successfully applied to establish the log-convexity of sequences of secondary structures [23], sequences counting directed column-convex polyominos [20], derangements, and certain classes of symmetric matrices with integer entries [22].…”

“…Log-convex functions are also important in communication engineering [12], pharmacokinetics [54,55], and atomic physics [3]. We conclude this non-exhaustive list by mentioning the appearance of log-convex sequences in mathematical biology, where they arise as the enumerating sequences of secondary structures of certain biopolymers [23,41,49]. Hence, the logarithmic behavior of sequences and functions is relevant in many non-combinatorial contexts, and this implies that the methods for establishing certain type of log-behavior are also interesting across many disciplines.…”