arXiv subjects
Publications and source records attributed to Francois Bergeron.
We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of (m,n)-parking functions, and associated (q,t)-analogs.
We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the ring of symmetric functions.
In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime and k > 1.
In the 90's a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.
This paper's aim is to present recent combinatorial considerations on r-Dyck paths, r-Parking functions, and the r-Tamari lattices. Giving a better understanding of the combinatorics of these objects has become important in view of their (conjectural) role in the description of the graded character of the Sn-modules of bivariate and trivariate diagonal coinvariant spaces for the symmetric group.
For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of $r$.
The operator nabla, introduced by Garsia and the author, plays a crucial role in many aspect of the study of diagonal harmonics. Besides giving several new formulas involving this operator, we show how one is lead to representation theoretic explanations for conjectures about the effect of this operator on Schur functions.
This is an overview (in french) of the Theory of Species for a general audience. Basic notions are introduced in a non too technical manner, with an explanation of why should one approach the notion of discrete structures in this particular way.
We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we work out explicitly this classification for degree 3 polynomials, and other special families of polynomials. This work extends to the singular case similar considerations of Martin, Savitt, and Singer for non-singular basketballs.
We study properties of (bi-infinite) arrays having all adjacent $k\times k$ adjacent minors equal to one. If we further add the condition that all adjacent $(k-1)\times (k-1)$ minors be nonzero, then these arrays are necessarily of rank $k$. It follows that we can explicit construct all of them. Several nice properties are made apparent. In particular, we revisit, with this perspective, the notion of frieze patterns of Coxeter. This shed new light on their properties. A connexion is also established with the notion of $T$-systems of Statistical Physics.
We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.
We explore in this paper the spaces of common zeros of several deformations of Steenrod operators.
The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.
The product $s_μs_ν$ of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We ask when expressions of the form $s_λs_ρ- s_μs_ν$ are Schur-positive. This general question seems to be a difficult one, but a conjecture of Fomin, Fulton, Li and Poon says that it is the case at least when $λ$ and $ρ$ are obtained from $μ$ and $ν$ by redistributing the parts of $μ$ and $ν$ in a specific, yet natural, way. We show that their conjecture is true in several significant cases. We also formulate a skew-shape extension of their conjecture, and prove several results which serve as evidence in favor of this extension. Finally, we take a more global view by studying two classes of partially ordered sets suggested by these questions.
The purpose of this paper is to give an explicit description of the irreducible decomposition of the multigraded S_n-module of coinvariants of S_n x S_n. Many of the results presented can be extended to analogous questions for other finite reflection group.
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.
The space $M_{μ/i,j}$ spanned by all partial derivatives of the lattice polynomial $Δ_{μ/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component $M_{μ/i,j}^0$ of $M_{μ/i,j}$. In particular, we give an explicit bases for $M_{μ/i,j}^0$ which allow us to prove directly the central {\sl four term recurrence} for these spaces.