Combinatorial studies of Euler's table on wreath products

Abstract : In the last two decades, much effort has been made to extend various enumerative results on symmetric groups to other more general groups. The main objective of this thesis is to extend to wreath products the results that concern the Euler's difference table. It is divided into five chapters. Euler's difference table associated to the sequence {n!} leads naturally to the counting formula for the derangements. In the first two chapters, we study Euler's difference table associated to the sequence {rnn!} and the generalized derangement problem. For the coefficients appearing in the later table, we give the combinatorial interpretations in terms of k-successions on wreath products. Clarke et al. introduced a q-analogue of Euler's difference table on symmetric group. In the third chapter, we extend their results to wreath products. By generalizing their bijection, we prove the equidistribution of the triple statistics “(fix, exc, fmaj)” and “(fix, exc, fmaf)” on wreath products, where “fmaf” is a new mahonian statistic on wreath products. On the other hand, Foata and Han have recently constructed two new transformations. We prove in fourth chapter that their two bijections provide a factorization of Clarke et al.'s bijection. In the fifth chapter we give an extension of Foata’s second fundamental transformation on r-colored words. We show that the bistatistics “(fmaj , des*)” and “(finv , col)” are equidistributed on wreath products, where “col” is the sum of color and “des*” a new statistic.
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General Mathematics [math.GM]. Université Claude Bernard - Lyon I, 2010. English. <NNT : 2010LYO10039>


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Hilarion Faliharimalala. Combinatorial studies of Euler's table on wreath products. General Mathematics [math.GM]. Université Claude Bernard - Lyon I, 2010. English. <NNT : 2010LYO10039>. <tel-00702743>

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