A combinatorial formula for LLT cumulants of melting lollipops in terms of spanning trees

We prove a combinatorial formula for LLT cumulants of melting lollipops as a positive combination of LLT polynomials indexed by spanning trees. The result gives an affirmative answer to a general positivity question for this class of unicellular LLT cumulants, and gives an independent proof of their Schur-positivity. In the special case of the complete graph, we also express the formula in terms of parking functions.Comment: An extended abstract of this work with fewer results and a different title is available at arXiv:2011.15080v

An instance of umbral methods in representation theory: the parking function module

We test the umbral methods introduced by Rota and Taylor within the theory of representation of symmetric group. We define a simple bijection between the set of all parking functions of length $n$ and the set of all noncrossing partitions of $\{1,2,...,n\}$. Then we give an umbral expression of the Frobenius characteristic of the parking function module introduced by Haiman that allows an explicit relation between this symmetric function and the volume polynomial of Pitman and Stanley

Free cumulants, Schr\"oder trees, and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.Comment: 23 page

A Hopf algebra of parking functions

If the moments of a probability measure on $\R$ are interpreted as a specialization of complete homogeneous symmetric functions, its free cumulants are, up to sign, the corresponding specializations of a sequence of Schur positive symmetric functions $(f_n)$. We prove that $(f_n)$ is the Frobenius characteristic of the natural permutation representation of \SG_n on the set of prime parking functions. This observation leads us to the construction of a Hopf algebra of parking functions, which we study in some detail.Comment: AmsLatex, 14 page

Positivity results for Stanley's character polynomials

Stanley introduced expressions for the normalized characters of the symmetric group and stated some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results by Goulden and the present author. We shall see that the generating series $C(t)$ introduced by them is critical to our discussion.Comment: 20 pages, 2 figures, v2, minor revisions, fixed typos et

Bell polynomials in combinatorial Hopf algebras

Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a first time, we show that most of the results on Bell polynomials can be written in terms of symmetric functions and transformations of alphabets. Then, we show that these results are clearer when stated in other Hopf algebras (this means that the combinatorial objects appear explicitly in the formulae). We investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the Lagrange-B{\"u}rmann formula

Random matrices, large deviations and reflected Brownian motion

In this thesis we present results in large deviations theory, free probability and the theory of reflected Brownian motion. We study the large deviations behaviour of the block structure of a non-crossing partition chosen uniformly at random. This allows us to apply the free momentcumulant formula of Speicher to express the spectral radius of a non-commutative random variable in terms of its free cumulants. Next the distributions of three quadratic functionals of the free Brownian bridge are studied: the square norm, the signature and the Lévy area of the free Brownian bridge. We introduce two representation of the free Brownian bridge as series involving free semicircular variables, analogous to classical results due to Lévy and Kac. The latter representation extends to all semicircular processes. For each of the three quadratic functionals we give the R-transform, from which we extract information about the distribution, including free infinite divisibility and smoothness of the density. We also apply our result about the spectral radius to compute the maximum of the support for Lévy area and square norm. In both cases we obtain implicit equations. The final chapter of the thesis is devoted to the study of a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This is motivated by recent developments in the theory of directed polymer and percolation models, in which existence of an invariant measure in product form plays a role. Informally, RBM is defined by running a standard Brownian motion in the polyhedral domain and giving it a singular drift whenever it hits one of the boundaries, kicking the process back into the interior. Our process is obtained by replacing this singular drift by a continuous one, involving a continuous potential. RBM has an invariant measure in product form if and only if a certain skew-symmetry condition holds. We show that this result extends to our generalisation. Applications include examples motivated by queueing theory, Brownian motion with rank-dependent drift and a process with close connections to the δ-Bose gas