Strict Log-Subadditivity for Overpartition Rank

Rank and Congruences for Overpartition Pairs

The rank of an overpartition pair is a generalization of Dyson’s rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of pp(n), the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.

Rank and Conjugation for the Frobenius Representation of an Overpartition

We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with resp...

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0, 1, ∗} is obtained by setting all ∗-entries to constants 0 or 1. A system of semi-linear equations over GF2 has the form Mx = f(x), where M is a completion of A and f : {0, 1}n → {0, 1}m is an operator, the ith coordinate of which can only depend on variables corresponding to ∗-entries in the ith row of A. We conjecture that no such system c...

Log-rank Test for Two Groups

Rank functions of strict cg-matroids

A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by S. Fujishige, G. A. Koshevoy, and Y. Sano in [9]. A cg-matroid whose rank function is naturally defined is called a strict cg-matroid. In this paper, we give characterizations of strict cg-matroids by their rank functions.