Which row of numbers has the most negatives — and why won't it change?

Mathematicians who study "shuffling 5 (or 50, or 500) objects" build a big grid of numbers out of it — some positive, some negative, some zero. One unsolved question asks: is there always one row that has more negative numbers in it than every other row? No calculator required — just some shape-drawing and pattern-spotting. Let's build it up from zero.