6533b862fe1ef96bd12c6dd9
RESEARCH PRODUCT
Statistics-preserving bijections between classical and cyclic permutations
Jean-luc Barilsubject
0102 computer and information sciencesFixed point[ MATH.MATH-CO ] Mathematics [math]/Combinatorics [math.CO]01 natural sciencesCombinatorial problemsTheoretical Computer ScienceCyclic permutationSet (abstract data type)CombinatoricsBijections[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]0101 mathematicsComputingMilieux_MISCELLANEOUSMathematicsDescent (mathematics)Discrete mathematicsStatistics on permutationsMathematics::Combinatorics010102 general mathematicsDescentComputer Science ApplicationsDerangement010201 computation theory & mathematicsExcedenceSignal ProcessingBijectionBijection injection and surjectionMaximaInformation Systemsdescription
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 step the set of weak excedances. Finally, some consequences and open problems are presented.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2012-10-16 |