Some asymptotic bijections

The notion of an asymptotic bijection is introduced and used to give bijective proofs of infinite summation formulas for set partitions (Dobinski's formula) and involutions.     

Partition identities and geometric bijections

We present a geometric framework for a class of partition identities. We show that there exists a unique bijection proving these identities, which satisfies certain linearity conditions. In particular, we show that Corteel's bijection enumerating partitions with nonnegative -th differences can be obtained by our approach. Other examples and generalizations are presented.

     

Memorandum on multiplicative bijections and order

Let denote the semigroup of continuous functions from the topological space X to , equipped with the pointwise multiplication. The paper studies semigroup homomorphisms , with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection has the form Tf=f○τ, where τ:X→Y is a Lipschitz homeomorphism. F. Cabello Sánchez and J. Cabello Sánchez are supported in part by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02. J. Cabello Sánchez is supported in part by a grant of the UEx (Programa Propio–Acción 2).     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

On bijections for pattern-avoiding permutations

By considering bijections from the set of Dyck paths of length 2n onto each of Sn(321) and Sn(132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207-219] gave a bijection that preserves the number of fixed points and the number of excedances in each σ∈Sn(321). We show that a direct bijection Γ:Sn(321)→Sn(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ. We also show that a bijection ?^∗:Sn(213)→Sn(321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133-148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383-409] preserves these same statistics, and we show that an analogous bijection from Sn(132) onto Sn(213) does the same.     

Partition bijections, a survey

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. 2000 Mathematics Subject Classification Primary—05A17; Secondary—05A30; 11P83 The author was partially supported by NSA and NSF grants.     

Tiling bijections between paths and Brauer diagrams

There is a natural bijection between Dyck paths and basis diagrams of the Temperley–Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.     

Polynomials hardly commuting with increasing bijections

Let I be an interval in the real line ℝ. Among the real polynomials that take I to I, we ask which ones do not commute with any increasing bijection of I other than identity. For this purely algebraic problem, the solution involves concepts in topological dynamics. Our main characterizations are in terms of full orbits of critical points and periodic points. Using these, we obtain simpler criterion, namely, that for no nontrivial subinterval K⊂I, the successive images {f ⁿ (K):n=0,1,2,…} form a pairwise disjoint collection. This problem is of interest in topological dynamics because it is about characterization of polynomials with unique self-topological-conjugacy.     

Additive symmetric bijections on geometric structures

Motivated by their existence in the algebraic theory of quadratic forms we define and study additive symmetric bijections on geometric structures. We give a construction of these bijections on IP-designs and then show that their existence on a design is equivalent to the design arising as the points and hyperplanes of a projective geometry. Using additive symmetric bijections we then develop a combinatorial framework for the study of finitely generated Witt rings.     

On bijections that preserve complementarity of subspaces

The set G of all m-dimensional subspaces of a 2m-dimensional vector space V is endowed with two relations, complementarity and adjacency. We consider bijections from G onto G^′, where G^′ arises from a 2m^′-dimensional vector space V^′. If such a bijection ? and its inverse leave one of the relations from above invariant, then also the other. In case m?2 this yields that ? is induced by a semilinear bijection from V or from the dual space of V onto V^′.As far as possible, we include also the infinite-dimensional case into our considerations.     

On actually computable bijections between ℕ and Q +

We present three different ways of getting an actually computable enumeration of Q ⁺ in the sense of being able to know exactly which rational occupies a given position and vice versa. The first enumeration is based on the Pierce expansion model for representing real numbers. The other two are based on regular continued fractions.The first and third authors were supported by UPF Grant, Support d'Iniciació a la Recerca, # F3087613.     

Linear bijections preserving the Hölder seminorm

Let be a compact metric space and let be a real number with The aim of this paper is to solve a linear preserver problem on the Banach algebra of Hölder functions of order from into We show that each linear bijection having the property that for every where

is of the form for every where with is a surjective isometry and is a linear functional.

     

关于正整数有序分拆的两个组合双射

研究了正整数有序分拆恒等式的组合证明.利用正整数有序分拆的共轭给出了与正整数有序分拆相关的两个组合双射.