Search results for "Combinatorics"
showing 10 items of 1770 documents
Motzkin subposets and Motzkin geodesics in Tamari lattices
2014
The Tamari lattice of order n can be defined by the set D n of Dyck words endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we study this rotation on the restricted set of Motzkin words. An upper semimodular join semilattice is obtained and a shortest path metric can be defined. We compute the corresponding distance between two Motzkin words in this structure. This distance can also be interpreted as the length of a geodesic between these Motzkin words in a Tamari lattice. So, a new upper bound is obtained for the classical rotation distance between two Motzkin words in a Tamari lattice. For some specific pairs of Motzkin words, this b…
Gray code for permutations with a fixed number of cycles
2007
AbstractWe give the first Gray code for the set of n-length permutations with a given number of cycles. In this code, each permutation is transformed into its successor by a product with a cycle of length three, which is optimal. If we represent each permutation by its transposition array then the obtained list still remains a Gray code and this allows us to construct a constant amortized time (CAT) algorithm for generating these codes. Also, Gray code and generating algorithm for n-length permutations with fixed number of left-to-right minima are discussed.
On the Non-uniform Redundancy of Representations for Grammatical Evolution: The Influence of Grammars
2018
The representation used in grammatical evolution (GE) is non-uniformly redundant as some phenotypes are represented by more genotypes than others. This article studies how the non-uniform redundancy of the GE representation depends on various types of grammars. When constructing the phenotype tree from a genotype, the used grammar determines Bavg, the average branching factor. Bavg measures the expected number of non-terminals chosen when mapping one genotype codon to a phenotype tree node. First, the paper illustrates that the GE representation induces a bias towards small trees. This bias gets stronger with lower Bavg. For example, when using a grammar with Bavg = 0.5, 75% of all genotype…
Radio Labelings of Distance Graphs
2013
A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.
A trace partitioned Gray code forq-ary generalized Fibonacci strings
2015
AbstractWe provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.
Two Reflected Gray Code-Based Orders on Some Restricted Growth Sequences
2014
We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant and ascent sequences, restricted growth functions and staircase words. In particular, we give the first suffix partitioned Gray codes for restricted growth f unctions and ascent sequences; these latter sequences code various combinatorial classes as interval orders, upper triangular matrices without zero rows and zero columns whose non-negative integer entries sum up to n, and certain pattern-avoiding permutations. For each Gr…
Metric Lie groups admitting dilations
2019
We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie …
On a class of generalised Schmidt groups
2015
In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An A 5 -free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.
Some new Hadamard designs with 79 points admitting automorphisms of order 13 and 19
2001
Abstract We have proved that there exists at least 2091 mutually nonisomorphic symmetric (79,39,19)-designs. In particular, 1896 of them admit an action of the nonabelian group of order 57, and an additional 194 an action of the nonabelian group of order 39.
The probability that $x$ and $y$ commute in a compact group
2010
We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…