A counting problem

As the tagline for this blog says, I like counting things. Reading my Iran diary reminded me of a counting problem I solved then, of which I am quite proud. But like all good problems, it leaves a loose end, which you might like to try. THe problem came from Charles Johnson, one offour pairs of my coauthhors with the same surname. He was interested in properties of a matrix which are at least partly controlled by its sign pattern, and wanted to count certain sign patterns. So a sign pattern is..