4,798 research outputs found
The KOH terms and classes of unimodal N-modular diagrams
We show how certain suitably modified N-modular diagrams of integer
partitions provide a nice combinatorial interpretation for the general term of
Zeilberger's KOH identity. This identity is the reformulation of O'Hara's
famous proof of the unimodality of the Gaussian polynomial as a combinatorial
identity. In particular, we determine, using different bijections, two main
natural classes of modular diagrams of partitions with bounded parts and
length, having the KOH terms as their generating functions. One of our results
greatly extends recent theorems of J. Quinn et al., which presented striking
applications to quantum physics.Comment: Several mostly minor or notational changes with respect to the first
version, in response to the referees' comments. 13 pages, 3 figures. To
appear in JCT
Odd values of the Klein j-function and the cubic partition function
In this note, using entirely algebraic or elementary methods, we determine a
new asymptotic lower bound for the number of odd values of one of the most
important modular functions in number theory, the Klein -function. Namely,
we show that the number of integers such that the Klein -function
--- or equivalently, the cubic partition function --- is odd is at least of the
order of for large. This improves
recent results of Berndt-Yee-Zaharescu and Chen-Lin, and approaches
significantly the best lower bound currently known for the ordinary partition
function, obtained using the theory of modular forms. Unlike many works in this
area, our techniques to show the above result, that have in part been inspired
by some recent ideas of P. Monsky on quadratic representations, do not involve
the use of modular forms.
Then, in the second part of the article, we show how to employ modular forms
in order to slightly refine our bound. In fact, our brief argument, which
combines a recent result of J.-L. Nicolas and J.-P. Serre with a classical
theorem of J.-P. Serre on the asymptotics of the Fourier coefficients of
certain level 1 modular forms, will more generally apply to provide a lower
bound for the number of odd values of any positive power of the generating
function of the partition function.Comment: A few minor revisions in response to the referees' comments. To
appear in the J. of Number Theor
Extending the idea of compressed algebra to arbitrary socle-vectors, II: cases of non-existence
This paper is the continuation of the previous work on generalized compressed
algebras (GCA's). First we exhibit a new class of socle-vectors which admit
a GCA (whose -vector is lower than the upper-bound of Theorem A of the
previous paper). In particular, it follows that for every socle-vector of
type 2 there exists a GCA (in any codimension ). The main result of this
paper is the following: there exist pairs which do not admit a GCA.
Moreover, the way this pathology occurs may be "arbitrarily bad" (even in
codimension 3). Finally, we start considering the difficult problem of
characterizing the pairs which admit a GCA, focusing on a particular
class of socle-vectors of codimension 3.Comment: 34 pages (19 in the journal). With permission from Elsevie
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