Effective scalar products of D-finite symmetric functions

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential equations, these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel's class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

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 numbers (Shareshian and Wachs, 2010 [17]). 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, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

Multiplicity Free Expansions of Schur <Emphasis Type="Italic">P</Emphasis>-Functions

After deriving inequalities on coefficients arising in the expansion of a Schur P-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free. Received February 28, 2005     

Symmetric functions and the centre of the Ringel-Hall algebra of a cyclic quiver

We obtain explicit generators for the centre of the Ringel-Hall algebra of a cyclic quiver and define a canonical algebra monomorphism from Macdonald's ring of symmetric functions to the centre, which furthermore respects the comultiplication and the symmetric bilinear form. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday     

Schur function identities, their t-analogs, and k-Schur irreducibility

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act trivially on the k-Schur functions, thus leading to a concept of irreducibility for these functions.     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

A Criterion for Bases of the Ring of Symmetric Functions

We give a criterion for bases of the ring of symmetric functions in n indeterminates over a commutative ring R with identity. A related algorithm is presented in the last section. Received April 13, 2004     

Descents, inversions, and major indices in permutation groups

A multivariate generating function involving the descent, major index, and inversion statistic first given by Ira Gessel is generalized to other permutation groups. We provide generating functions for variants of these three statistics for the Weyl groups of type B and D, wreath product groups, and multiples of permutations. All of our ideas are combinatorial in nature and exploit fundamental relationships between the elementary and homogeneous symmetric functions.     

On the unimodality of spherically symmetric stable distribution functions

Several theorems are obtained concerning the unimodality of spherically symmetric distribution functions. These theorems are used to show that a class of spherically symmetric infinitely divisible distribution functions that contains the class of spherically symmetric stable distribution functions is unimodal.     

Mould calculus,polyhedral cones,and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation.     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.     

Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients.Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.     

Noncommutative symmetric functions and an amazing matrix

We present a simple way to derive the results of Diaconis and Fulman [P. Diaconis, J. Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, arXiv:1102.5159] in terms of noncommutative symmetric functions.     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

A basic class of symmetric orthogonal functions with six free parameters

In a previous paper, we introduced a basic class of symmetric orthogonal functions (BCSOF) by an extended theorem for Sturm-Liouville problems with symmetric solutions. We showed that the foresaid class satisfies the differential equation

     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

A symmetric generalization of Sturm–Liouville problems in q-difference spaces

Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.     

Orthogonality of Jack polynomials in superspace

Jack polynomials in superspace, orthogonal with respect to a “combinatorial” scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an “analytical” scalar product, introduced in [P. Desrosiers, L. Lapointe, P. Mathieu, Jack polynomials in superspace, Comm. Math. Phys. 242 (2003) 331-360] as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in [P. Desrosiers, L. Lapointe, P. Mathieu, Classical symmetric functions in superspace, J. Algebraic Combin. 24 (2006) 209-238].     

Decomposition of submodular functions

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.