The original Macdonald polynomials P_μ form a basis for the vector space of symmetric functions which specializes to several of the common bases such as the monomial, Schur, and elementary bases. There are a number of different types of Macdonald polynomials obtained from the original P_μ through a combination of algebraic and plethystic transformations one of which is the modified Macdonald polynomial H̃_μ. In this dissertation, we study a certain specialization F̃_μ(q,t) which is the coefficient of x₁x₂…x_N in H̃_μ and also the Hilbert series of the Garsia-Haiman module M_μ. Haglund found a combinatorial formula expressing F̃_μ as a sum of n! objects weighted by two statistics. Using this formula we prove a q,t-analogue of the hook-length formula for hook shapes. We establish several new combinatorial operations on the fillings which generate F̃_μ. These operations are used to prove a series of recursions and divisibility properties for F̃_μ. / Ph. D.

In this paper, new variations of some well-known permutation statistics are introduced and studied. Firstly, a double-descent of a permutation π is defined as a position i where πi ≥ 2πi+1. By proofs by induction and direct proofs, recursive and explicit expressions for the number of n-permutations with k double-descents are presented. Also, an expression for the total number of double-descents in all n-permutations is presented. Secondly, a double-inversion of a permutation π is defined as a pair (πi,πj) where i<j but πi ≥ 2πj. The total number of double-inversions in all n-permutations is presented.

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler's exponential generating function formula for the Eulerian polynomials. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics. The family of colored permutation groups includes the family of symmetric groups and the family of hyperoctahedral groups, also called the type A Coxeter groups and type B Coxeter groups, respectively. By specializing our formulas to these cases, they reduce to the Shareshian-Wachs q-analog of Euler's formula, formulas of Foata and Han, and a new generalization of a formula of Chow and Gessel.

In 2010 Chung-Graham-Knuth proved an interesting symmetric identity for the Eulerian numbers and asked for a q-analog version. Using the q-Eulerian polynomials introduced by Shareshian-Wachs we find such a q-identity. Moreover, we provide a bijective proof that we further generalize to prove other symmetric qidentities using a combinatorial model due to Foata-Han. Meanwhile, Hyatt has introduced the colored Eulerian quasisymmetric functions to study the joint distribution of the excedance number and major index on colored permutations. Using the Decrease Value Theorem of Foata-Han we give a new proof of his main generating function formula for the colored Eulerian quasisymmetric functions. Furthermore, certain symmetric q-Eulerian identities are generalized and expressed as identities involving the colored Eulerian quasisymmetric functions. Next, generalizing the recent works of Savage-Visontai and Beck-Braun we investigate some q-descent polynomials of general signed multipermutations. The factorial and multivariate generating functions for these q-descent polynomials are obtained and the real rootedness results of some of these polynomials are given. Finally, we study the diagonal generating function of the Jacobi-Stirling numbers of the second kind by generalizing the analogous results for the Stirling and Legendre-Stirling numbers of the second kind. It turns out that the generating function is a rational function, whose numerator is a polynomial with nonnegative integral coefficients. By applying Stanley's theory of P-partitions we find combinatorial interpretations of those coefficients

En 2010 Chung, Graham et Knuth ont démontré une remarquable identité symétrique sur les nombres eulériens et posé le problème de trouver un q-analogue de leur identité. En utilisant les q-polynômes eulériens introduits par Shareshian-Wachs, nous avons pu obtenir une telle q-identité. La preuve bijective que nous avons imaginée, nous a permis ensuite de démontrer d'autres q-identités symétriques, en utilisant un modèle combinatoire dû à Foata-Han. Entre temps, Hyatt a introduit les fonctions quasisymétriques eulériennes colorées afin d'étudier la distribution conjointe du nombre d'excédances et de l'indice majeur sur les permutations colorées. En appliquant le Decrease Value Theorem de Foata-Han, nous donnons d'abord une nouvelle preuve de sa formule principale sur la fonction génératrice des fonctions quasisymétriques eulériennes colorées, puis généralisons certaines identités eulériennes symétriques, en les exprimant comme des identités sur les fonctions quasisymétriques eulériennes colorées. D'autre part, en prolongeant les travaux récents de Savage-Visontai et Bec-raun, nous considérons plusieurs q-polynômes de descente des mots signés. Leurs fonctions génératrices factorielles et multivariées sont explicitement calculées. Par ailleurs, nous montrons que certains de ces polynômes n'ont que des zéros réels. Enfin, nous étudions la fonction génératrice diagonale des nombres de Jacobi Stirling de deuxième espèce, en généralisant des résultats analogues pour les nombres de Stirling et Legendre-Stirling de deuxième espèce. Il s'avère que cette fonction génératrice est une série rationnelle dont le numérateur est un polynôme à coefficients entiers positifs. En appliquant la théorie des P-partitions de Stanley nous trouvons des interprétations combinatoires de ces coefficients / In 2010 Chung-Graham-Knuth proved an interesting symmetric identity for the Eulerian numbers and asked for a q-analog version. Using the q-Eulerian polynomials introduced by Shareshian-Wachs we find such a q-identity. Moreover, we provide a bijective proof that we further generalize to prove other symmetric qidentities using a combinatorial model due to Foata-Han. Meanwhile, Hyatt has introduced the colored Eulerian quasisymmetric functions to study the joint distribution of the excedance number and major index on colored permutations. Using the Decrease Value Theorem of Foata-Han we give a new proof of his main generating function formula for the colored Eulerian quasisymmetric functions. Furthermore, certain symmetric q-Eulerian identities are generalized and expressed as identities involving the colored Eulerian quasisymmetric functions. Next, generalizing the recent works of Savage-Visontai and Beck-Braun we investigate some q-descent polynomials of general signed multipermutations. The factorial and multivariate generating functions for these q-descent polynomials are obtained and the real rootedness results of some of these polynomials are given. Finally, we study the diagonal generating function of the Jacobi-Stirling numbers of the second kind by generalizing the analogous results for the Stirling and Legendre-Stirling numbers of the second kind. It turns out that the generating function is a rational function, whose numerator is a polynomial with nonnegative integral coefficients. By applying Stanley’s theory of P-partitions we find combinatorial interpretations of those coefficients