Dragutin Svrtan

Enumerative aspects of secondary structures

Abstract A secondary structure is a planar, labeled graph on the vertex set {1,…, n } having two kind of edges: the segments [ i , i +1], for 1⩽ i ⩽ n −1 and arcs in the upper half-plane connecting some vertices i , j , i ⩽ j , where j − i > l , for some fixed integer l. Any two arcs must be totally disjoint. We enumerate secondary structures with respect to their size n, rank l and order k (number of arcs), obtaining recursions and, in some cases, explicit formulae in terms of Motzkin, Catalan, and Narayana numbers. We give the asymptotics for the enumerating sequences and prove their log-convexity, log-concavity and unimodality. It is shown how these structures are connected with hypergeometric functions and orthogonal polynomials.

Exponential Formulas and Lie Algebra Type Star Products

Given formal differential operators Fi on polynomial algebra in several variables x1;:::;xn, we discuss finding expressions Kl determined by the equation exp( P i xiFi)(exp( P j qjxj)) = exp( P l Klxl) and their applications. The expressions for Kl are related to the coproducts for deformed momenta for the noncommutative space-times of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding Kl. We elaborate an example for a Lie algebra su(2), related to a quantum gravity application from the literature.

Combinatorics of diagonally convex directed polyominoes

Abstract A new bijection between the diagonally convex directed (dcd-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raneys generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomino can grow at most by one, the problem of q -enumeration of this object can be solved by an application of Gessels q -analog of the Lagrange inversion formula.

Partition functions for general multi-level systems

Abstract We describe a unified approach for calculating the partition functions of a general multi-level system with a free Hamiltonian. Particularly, we present new results for parastatistical systems of any order in the second quantized approach. Anyonic-like systems are briefly discussed.

New plethysm operation, Chern characters of exterior and symmetric powers with applications to Stiefel-Whitney classes of grassmannians

Abstract In this paper a new plethysm operation is proposed and a technique for coefficient extraction for a fairly general class of symmetric power series (e.g. multiplicative sequences of the theory of ?characteristic classes) is developed, together with various applications.