19 research outputs found
Exponential Formulas and Lie Algebra Type Star Products
Given formal differential operators on polynomial algebra in several
variables , we discuss finding expressions determined by the
equation
and their applications. The expressions for 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
. We elaborate an example for a Lie algebra , related to a quantum
gravity application from the literature
New identities for the polarized partitions and partitions with -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
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
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