Études combinatoires des nombres de Jacobi-Stirling et d'Entringer

Abstract : This thesis is constructed in two main independant parts ; the first one dealing with the numbers of Jacobi-Stirling, the second one tackling the numbers of Entringer. The first part introduces the numbers of Jacobi-Stirling of the second kind and of the first kind, as algebraic coefficients in some polynomial relations. We give some combinatorial interpretations of these numbers, in terms of set partitions and quasi-permutations for the numbers of the second kind, and in terms of permutations for the numbers of the first kind. We also study the diagonal generating functions of these sequences of numbers, and one of their generalization based on the model of r-Stirling numbers. The second part introduces the numbers of Entringer with their interpretation in terms of alternating permutations. We study the different recurrences formulas satisfied by these numbers, and refine these results with a q-analogue using the inversion statistic. We also note that these results can be extend to permutations with any fixed shape. Finally, we define the notion of Entringer family, and provide bijections between some of these families. In particular, we establish a bijection between the alternating permutations with fixed given value, and the binary increasing trees such that the end-point of the minimal path is fixed.
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Mathematics. Université Claude Bernard - Lyon I, 2010. French. <NNT : 2010LYO10156>
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Yoann Gelineau. Études combinatoires des nombres de Jacobi-Stirling et d'Entringer. Mathematics. Université Claude Bernard - Lyon I, 2010. French. <NNT : 2010LYO10156>. <tel-00531200v2>

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