We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le f([n_1,\ldots,n_k])$ involving gcd lcm of integers $n_1,\ldots,n_k$, where $f$ belongs to certain classes additive arithmetic functions. In particular, we consider generalized omega function $\Omega_{\ell}(n)= \sum_{p^\nu \mid\mid n} \nu^{\ell}$ investigated by Duncan (1962)...

In this section we will discuss the Inclusion-Exclusion principle, with a few applications (including a formula for the chromatic polynomial of a graph), and then consider a wide generalisation of it due to Gian-Carlo Rota, involving the Möbius function of a partially ordered set. The q-binomial theorem gives a simple formula for the Möbius function of the lattice of subspaces of a vector space.

Optical time division multiplexing (OTDM) allows multiple traffic streams to share the bandwidth of a wavelength efficiently. In this paper, we present a new analytical model, based on the inclusion-exclusion principle from combinatorics, for evaluating the blocking performance of time-space switched optical networks with fixed routing and random wavelength/timeslot assignment. This model can b...

We present an information-theoretic framework for mining dependencies between itemsets in binary data. The problem of closure-based redundancy in this context is theoretically investigated, and we present both lossless and lossy pruning techniques. An efficient and scalable algorithm is proposed, which exploits the inclusion-exclusion principle for fast entropy computation. This algorithm is em...

‘Magnitude’ is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion pr...

By introducing the notion of relative derangements of type B, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type B analogue of the well-known relation between the relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretatio...