Notation: In the sequel K is a field and Σn is the group of permutation over n elements. If A is an operad, then the K-vector space of n-ary operations is denoted as usual by A(n). Recall that if A is regular, then A(n) := An⊗KΣn, where An is the K-vector space of n-ary operations without permutations of the entries. We adopt Sweedler notation for binary cooperation ∆ on a K-vector space V and ...

We characterize the set of all n-ary quasigroups of order 4: every n-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an n-ary quasigroup can be represented as a composition of k-ary and (n− k +1)-ary quasigroups for some k from 2 to n−1, where the order of arguments in the representation can differ from the original order. The set of semilinea...

This article investigates the problem of designing virtual di-paths (VPs) in a directed ATM model, in which the ow of information in the two directions of a link are not identical. On top of a given physical network we construct directed VPs. Routing in the physical network is done using these VPs. Given the capacity of each physical link (the maximum number of VPs that can pass through the lin...

In Bevan et al. (2016), the study of patterns in forests of binary shrubs was introduced. A k-ary heap H is a k-ary tree labeled with {1, . . . , n} such that every child has a larger label than its parent. Given a k-ary heap H , we associate a permutation σH with H by recording the vertex labels as they are encountered in the breadth-first search of the tree. For example, in Figure 1, we pictu...

In this paper we investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. Our methods involve a generalization of the Jordan formalization of observables in quantum mechanics in n-ary splicing algebras. We show that the splicing algebras are an n-ary envelope for algebras of DNA recombinati...

The n-ary first and second recursion theorems formalize two distinct, yet similar, notions of self-reference. Roughly, the n-ary first recursion theorem says that, for any n algorithmic tasks (of an appropriate type), there exist n partial computable functions that use their own graphs in the manner prescribed by those tasks; the n-ary second recursion theorem says that, for any n algorithmic t...

The edge-bandwidth problem is an analog of the classical bandwidth problem, in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized. We prove tight bounds on the edge-bandwidth of hypercube and butterfly graphs and complete k-ary trees which extend and improve on previous known results. We also provide...

In this paper we introduce a notion of divisibility and primality on k-ary trees and we nd a relation between indecomposable preex code and prime trees. This relation allows to work on trees instead than directly on preex codes. In this way we can prove a density theorem for indecomposable preex codes and we can give an algorithm to test the indecomposability of a maximal preex code.

In this paper, we present an involution on some kind of colored k-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers Ck,γ(n) = γ kn+γ ( kn+γ n ) . From the combinatorial sum, we refine the formula for k-ary trees and obtain an implicit formula for the generating function of the generalized Catalan numbers which obviously implies a Van...