On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions

Abstract: We prove a lower bound expansion on the probability that a random $\pm 1$ matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form $e_i \pm e_j$, which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form $e_i \pm e_j \pm e_k \pm e_\ell$, which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the "novel partitions" which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are "first place and runners up," in relation to the Erd\H{o}s-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.