6 research outputs found
Anti-lecture Hall Compositions and Overpartitions
We show that the number of anti-lecture hall compositions of n with the first
entry not exceeding k-2 equals the number of overpartitions of n with
non-overlined parts not congruent to modulo k. This identity can be
considered as a refined version of the anti-lecture hall theorem of Corteel and
Savage. To prove this result, we find two Rogers-Ramanujan type identities for
overpartition which are analogous to the Rogers-Ramanjan type identities due to
Andrews. When k is odd, we give an alternative proof by using a generalized
Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and
a refined version of a bijection also due to Corteel and Savage.Comment: 16 page
The Rogers-Ramanujan-Gordon Theorem for Overpartitions
Let be the number of partitions of with certain difference
condition and let be the number of partitions of with certain
congruence condition. The Rogers-Ramanujan-Gordon theorem states that
. Lovejoy obtained an overpartition analogue of the
Rogers-Ramanujan-Gordon theorem for the cases and . We find an
overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general
case. Let be the number of overpartitions of satisfying
certain difference condition and be the number of overpartitions
of whose non-overlined parts satisfy certain congruences condition. We show
that . By using a function introduced by Andrews, we
obtain a recurrence relation which implies that the generating function of
equals the generating function of . We also find a
generating function formula of by using Gordon marking
representations of overpartitions, which can be considered as an overpartition
analogue of an identity of Andrews for ordinary partitions.Comment: 26 page
Classical symmetric functions in superspace
We present the basic elements of a generalization of symmetric function
theory involving functions of commuting and anticommuting (Grassmannian)
variables. These new functions, called symmetric functions in superspace, are
invariant under the diagonal action of the symmetric group on the sets of
commuting and anticommuting variables. In this work, we present the superspace
extension of the classical bases, namely, the monomial symmetric functions, the
elementary symmetric functions, the completely symmetric functions, and the
power sums. Various basic results, such as the generating functions for the
multiplicative bases, Cauchy formulas, involution operations as well as the
combinatorial scalar product are also generalized.Comment: 21 pages, this supersedes the first part of math.CO/041230
Overpartitions and Bressoud's conjecture, I
In this paper, we introduce a new partition function which
could be viewed as an overpartition analogue of the partition function
introduced by Bressoud. By constructing a bijection, we showed that there is a
relationship between and and a relationship between
and . Based on the relationship between
and and Bressoud's theorems on the Rogers-Ramanujan-Gordon identities and
the G\"ollnitz-Gordon identities, we obtain the overpartition analogue of the
Rogers-Ramanujan-Gordon identities due to Chen, Sang and Shi and a new
overpartition analogue of the Andrews-G\"ollnitz-Gordon identities. On the
other hand, by using the relation between and and
Bressoud's conjecture for proved by Kim, we obtain an overpartition
analogue of Bressoud's conjecture for , which provides overpartition
analogues of many classical partition theorems including Euler's partition
theorem. The generating function of the overpartition analogue of Bressoud's
conjecture for is also obtained with the aid of Bailey pairs
Les superpolynômes de Jack et le modèle Calogero-Moser-Sutherland N = 2
Dans cet ouvrage, on présente une généralisation des polynômes symétriques de Jack, les superpolynômes de Jack N = 2, et on discute de ses connections avec le modèle Calogero-Moser-Sutherland trigonométrique (tCMS) supersymétrique N = 2. On fait d’abord une brève introduction à la théorie des polynômes symétriques pour ensuite définir le polynôme symétrique de Jack. On le définit de trois façons : combinatoirement, en tant que fonction propre du modèle tCMS et comme le résultat de la symétrisation du polynôme de Jack non symétrique. On introduit ensuite la théorie des superpolynômes symétriques N = 1. Le superpolynôme de Jack est alors défini selon les trois mêmes approches adaptées au superespace. On procède ensuite à la construction des superpolynômes N = 2 et à la construction du modèle tCMS avec deux supersymétries, à la suite de quoi les quantités conservées du modèle sont présentées. Ultimement, on pose une première définition des superpolynômes de Jack N = 2. On montre alors que ceux-ci sont les fonctions propres du modèle tCMS N = 2 et de ses quantités conservées. On obtient auxiliairement une définition combinatoire de ces superpolynômes qui est conjecturée équivalente à la première.We present a generalization of the symmetric Jack polynomial, the N = 2 symmetric Jack superpolynomial, and discuss its links with the N = 2 supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (tCMS) model. We first briefly review the theory of symmetric polynomials that leads us to three different definitions of the symmetric Jack polynomials: a combinatorial definition, the Jack polynomial as the eigenfunction of the tCMS model and as the result of the symmetrization of the non-symmetric Jack polynomial. We then do a brief introduction to the theory of symmetric superpolynomials. We also define the symmetric Jack superpolynomials using the superextension of the three aforementioned characterizations. After this introduction, we get to the main matter by defining the symmetric N = 2 superpolynomials. This ultimately results in a definition of the N = 2 Jack superpolynomial. We construct a N = 2 superextension of the tCMS model and find its conserved quantities. The N = 2 Jack superpolynomials are found to be the eigenfunctions of this model. As an auxiliary result, we obtain a conjecture regarding a combinatorial definition of these superpolynomials