585 research outputs found
Overpartition pairs and two classes of basic hypergeometric series
We study the combinatorics of two classes of basic hypergeometric series. We
first show that these series are the generating functions for certain
overpartition pairs defined by frequency conditions on the parts. We then show
that when specialized these series are also the generating functions for
overpartition pairs with bounded successive ranks, overpartition pairs with
conditions on their Durfee dissection, as well as certain lattice paths. When
further specialized, the series become infinite products, leading to numerous
identities for partitions, overpartitions, and overpartition pairs.Comment: 31 pages, To appear in Adv. Mat
Automorphic properties of generating functions for generalized rank moments and Durfee symbols
We define two-parameter generalizations of two combinatorial constructions of
Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We
prove that three specializations of the associated generating functions are
so-called quasimock theta functions, while a fourth specialization gives
quasimodular forms. We then define a two-parameter generalization of Andrews'
smallest parts function and note that this leads to quasimock theta functions
as well. The automorphic properties are deduced using q-series identities
relating the relevant generating functions to known mock theta functions.Comment: 18 page
Inequalities for full rank differences of 2-marked Durfee symbols
In this paper, we obtain infinitely many non-trivial identities and
inequalities between full rank differences for 2-marked Durfee symbols, a
generalization of partitions introduced by Andrews. A certain strict
inequality, which almost always holds, shows that identities for Dyson's rank,
similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By
showing an analogous strict inequality, we show that such non-trivial
identities are also rare for the full rank, but on the other hand we obtain an
infinite family of non-trivial identities, contrasting the partition theoretic
case.Comment: 21 page
Non-Abelian spin-singlet quantum Hall states: wave functions and quasihole state counting
We investigate a class of non-Abelian spin-singlet (NASS) quantum Hall
phases, proposed previously. The trial ground and quasihole excited states are
exact eigenstates of certain k+1-body interaction Hamiltonians. The k=1 cases
are the familiar Halperin Abelian spin-singlet states. We present closed-form
expressions for the many-body wave functions of the ground states, which for
k>1 were previously defined only in terms of correlators in specific conformal
field theories. The states contain clusters of k electrons, each cluster having
either all spins up, or all spins down. The ground states are non-degenerate,
while the quasihole excitations over these states show characteristic
degeneracies, which give rise to non-Abelian braid statistics. Using conformal
field theory methods, we derive counting rules that determine the degeneracies
in a spherical geometry. The results are checked against explicit numerical
diagonalization studies for small numbers of particles on the sphere.Comment: 17 pages, 4 figures, RevTe
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