Search results for "Bijection"

Symmetric and finitely symmetric polynomials on the spaces ℓ∞ and L∞[0,+∞)

2018

We consider on the space l∞ polynomials that are invariant regarding permutations of the sequence variable or regarding finite permutations. Accordingly, they are trivial or factor through c0. The analogous study, with analogous results, is carried out on L∞[0,+∞), replacing the permutations of N by measurable bijections of [0,+∞) that preserve the Lebesgue measure.

Statistics-preserving bijections between classical and cyclic permutations

2012

Recently, Elizalde (2011) [2] has presented a bijection between the set C"n"+"1 of cyclic permutations on {1,2,...,n+1} and the set of permutations on {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. In this paper, we construct a bijection from C"n"+"1 to S"n that preserves the weak excedance set and that transfers quasi-fixed points into fixed points and left-to-right maxima into themselves. This induces a bijection from the set D"n of derangements to the set C"n"+"1^q of cycles without quasi-fixed points that preserves the weak excedance set. Moreover, we exhibit a kind of discrete continuity between C"n"+"1 and S"n that preserves at each s…

Correspondences of Brauer characters and Sylow subgroup normalizers

2021

Abstract Let p > 3 and q ≠ p be primes, let G be a finite q-solvable group and let P ∈ Syl p ( G ) . Then G has a unique irreducible q-Brauer character of p ′ -degree lying over 1 P if and only if N G ( P ) / P is a q-group. One direction of this result follows from a natural McKay bijection of p ′ -degree irreducible q-Brauer characters, which is obtained under suitable conditions.

Correspondences Between 2-Brauer Characters of Solvable Groups

2010

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl p (G) and N = N G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N G (P).

A McKay bijection for projectors

2021

On bijections vs. unary functions

1996

A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively.

Orbits of bounded bijective operators and Gabor frames

2020

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(\mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $\mathbb{Z}$, which are orbits of bounded operators on $L^2(\mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $\mathbb{Z}$ and represented by orbits of operators in $GL(L^2(\mathbb{R}))$ are given. Some results about opera…

Combinatorial isomorphism between Fibonacci classes

2008

Abstract In 1985 Simion and Schmidt showed that the set S n (T 3) of length n permutations avoiding the set of patterns T 3={123, 132, 213} is counted by (the second order) Fibonacci numbers. They also presented a constructive bijection between the set F n–1 of length (n–1) binary strings with no two consecutive 1s and S n (T 3). In 2005, Egge and Mansour generalized the first Simion-Simion’s result and showed that S n (T p ), the set of permutations avoiding the patterns T p ={12…p, 132, 213}, is counted by the (p–1)th order Fibonacci numbers. In this paper we extend the second Simion-Schmidt’s result by giving a bijection between the set of length (n–1) binary strings with no (p–1) consec…

A bijection between words and multisets of necklaces

2012

Two of the present authors have given in 1993 a bijection Phi between words on a totally ordered alphabet and multisets of primitive necklaces. At the same time and independently, Burrows and Wheeler gave a data compression algorithm which turns out to be a particular case of the inverse of Phi. In the present article, we show that if one replaces in Phi the standard permutation of a word by the co-standard one (reading the word from right to left), then the inverse bijection is computed using the alternate lexicographic order (which is the order of real numbers given by continued fractions) on necklaces, instead of the lexicographic order as for Phi(-1). The image of the new bijection, ins…

Lehmer code transforms and Mahonian statistics on permutations

2012

Abstract In 2000 Babson and Steingrimsson introduced the notion of vincular patterns in permutations. They show that essentially all well-known Mahonian permutation statistics can be written as combinations of such patterns. Also, they proved and conjectured that other combinations of vincular patterns are still Mahonian. These conjectures were proved later: by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In this paper we give an alternative proof of some of these results. Our approach is based on permutation codes which, like the Lehmer code, map bijectively permutations onto subexcedant sequences. More precisely, we give several code transforms (i.e., bijections…