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 $j$-function. Namely, we show that the number of integers $n\le x$ such that the Klein $j$-function --- or equivalently, the cubic partition function --- is odd is at least of the order of $$\frac{\sqrt{x} \log \log x}{\log x},$$ for $x$ 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 $s$ which admit a GCA (whose $h$-vector is lower than the upper-bound $H$ of Theorem A of the previous paper). In particular, it follows that for every socle-vector $s$ of type 2 there exists a GCA (in any codimension $r$). The main result of this paper is the following: there exist pairs $(r,s)$ 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 $(r,s)$ 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