Exponential Formulas and Lie Algebra Type Star Products

Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,...,x_n$, we discuss finding expressions $K_l$ determined by the equation $\exp(\sum_i x_i F_i)(\exp(\sum_j q_j x_j)) = \exp(\sum_l K_l x_l)$ and their applications. The expressions for $K_l$ 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 $K_l$. We elaborate an example for a Lie algebra $su(2)$, related to a quantum gravity application from the literature

New identities for the polarized partitions and partitions with dd-distant parts

In this paper we present a new class of integer partition identities. The number of partitions with d-distant parts can be represented as a sum of the number of partitions with 1-distant parts whose even parts are greater than twice the number of odd parts. We also provide a direct bijection between these classes of partitions

Some Refinements of Formulae Involving Floor and Ceiling Functions

The floor and ceiling functions appear often in mathematics and manipulating sums involving floors and ceilings is a subtle game. Fortunately, the well-known textbook Concrete Mathematics provides a nice introduction with a number of techniques explained and a number of single or double sums treated as exercises. For two such double sums we provide their single-sum analogues. These closed-form identities are given in terms of a dual partition of the multiset (regarded as a partition) of all b-ary digits of a nonnegative integer. We also present the double- and single-sum analogues involving the fractional part function and the shifted fractional part function

Combinatorics of diagonally convex directed polyominoes

AbstractA 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 Raney's 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 Gessel's q-analog of the Lagrange inversion formula

Enumerative aspects of secondary structures

AbstractA 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