Thesis
French
ID: <
10670/1.oie1z7>
Abstract
This thesis is divided into two parts, the first deals with the combinatorics associated to the normal ordering form of noncommutative operators and the second addresses the symmetric distributions of the crossing numbers and nesting numbers, respectively k-crossings and k-nestings, in combinatorial structures (partitions, permutations, colored permutations, …). The first part studies the normal order of operators in terms of rook placements. We study the normal ordering form connecting two noncommutative operators D and U, and some special orthogonal polynomials, and establish bijonctions between coefficients of (D+U)n and rook placements in Ferrers diagrams. We also give combinatorial proofs and alternatives to some quantum conjectures posed by physicists. In the second part, we define the notions of statistics, nestings and k-nestings, on the sets of permutations of the Coxeter group of type B. We also give extensions to type B of the results of the crossings and nestings, respectivelu k-crossings and K-nestings in the set of permutations of type A, in terms of symmetric distributions. Likewise, we study the link between non-commutative operators and these statistics. Other extensions of the distribution of these statistics on the sets of colored partitions and colored permutations of type A and B are established