Search results for "COD"
showing 10 items of 2985 documents
A new Euler–Mahonian constructive bijection
2011
AbstractUsing generating functions, MacMahon proved in 1916 the remarkable fact that the major index has the same distribution as the inversion number for multiset permutations, and in 1968 Foata gave a constructive bijection proving MacMahon’s result. Since then, many refinements have been derived, consisting of adding new constraints or new statistics.Here we give a new simple constructive bijection between the set of permutations with a given number of inversions and those with a given major index. We introduce a new statistic, mix, related to the Lehmer code, and using our new bijection we show that the bistatistic (mix,INV) is Euler–Mahonian. Finally, we introduce the McMahon code for …
A loopless algorithm for generating the permutations of a multiset
2003
AbstractMany combinatorial structures can be constructed from simpler components. For example, a permutation can be constructed from cycles, or a Motzkin word from a Dyck word and a combination. In this paper we present a constructor for combinatorial structures, called shuffle on trajectories (defined previously in a non-combinatorial context), and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects.
The structure of the state representation of shift invariant controllable and observable group codes
2000
AbstractIn this paper an investigation on the structure of the canonical trellis section of shift invariant, l-controllable and m-observable group codes is carried out. Necessary and sufficient conditions for a set of group homomorphisms in order that they represent the trellis section of this class of codes are established.
A fractal set from the binary reflected Gray code
2005
The permutation associated with the decimal expression of the binary reflected Gray code with $N$ bits is considered. Its cycle structure is studied. Considered as a set of points, its self-similarity is pointed out. As a fractal, it is shown to be the attractor of a IFS. For large values of $N$ the set is examined from the point of view of time series analysis
Analysis of properties of recombination operators proposed for the node-depth encoding
2011
The node-depth encoding is a representation for evolutionary algorithms applied to tree problems. Its represents trees by storing the nodes and their depth in a proper ordered list. The original formulation of the node-depth encoding has only mutation operators as the search mechanism. Although it is computationally efficient, the exclusive use of mutation restricts the exploration of the search space and the algorithm convergence. Then, this work proposes two specific recombination operators to improve the convergence of the algorithm using the node-depth encoding representation. These operators are based on recombination operators for permutation representations. Analysis of the proposed …
Polynomial codimension growth and the Specht problem
2017
Abstract We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α ∈ ( 0 , 1 ) a commutative nonassociative algebra A α whose codimension sequence c n ( A α ) , n = 1 , 2 , … , is polynomially bounded and lim log n c n ( A α ) = 3 + α . As an application we are able to construct a new example of a variety with an infinite basis of identities.
Standard polynomials are characterized by their degree and exponent
2011
Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .
Polynomial identities on superalgebras: Classifying linear growth
2006
Abstract We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.
Codimension and colength sequences of algebras and growth phenomena
2015
We consider non necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe some of the results obtained in recent years on the sequence of codimensions and the sequence of colengths of an algebra.
On algebras of polynomial codimension growth
2016
Let A be an associative algebra over a field F of characteristic zero and let $$c_n(A), n=1, 2, \ldots $$ , be the sequence of codimensions of A. It is well-known that $$c_n(A), n=1, 2, \ldots $$ , cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.