The Anatomy of Integers and Permutations

If you switch on your TV in the evening then, as likely as not, you will find yourself watching an episode of a popular detective show (set in various spectacular locations) in which surprisingly dapper forensic scientists turn up evidence using careful anatomical (and other) study so as to be able to identify and prosecute a heinous criminal. Sometimes a flatfooted detective is misled by the surface evidence to suspect one person, but then the forensic team, digging deeper, turns up details that surprise not only the easily misled detective but even you, the astute viewer. For example, two seemingly unrelated corpses are found, and our hapless detective believes that the crimes are unrelated, whereas the forensic investigators turn up conclusive proof that the two corpses were in fact twins. So what would happen if we put together a forensic team to investigate the anatomy of some of the most common mathematical objects, say of integers and permutations? Seems silly at first. Most of our training with these simple mathematical objects involves how they are used in understanding more complicated phenomena, but rarely do we look at their anatomy, the inter-relation of their constituent parts (that is, the prime factors of integers, and the cycles of permutations). So our objective is to be the forensic scientists, with the corpses of these two seemingly unrelated mathematical objects laid out before us, and it is up to us to determine whether there is more in common between the anatomies of integers and permutations than meets the eye. This article is written as a companion piece to [0], a film-script in which we develop the connection with anatomy and forensics to create a fantasy world where forensic detectives (loosely based on famous mathematicians) prove and interpret several of the key notions exposed more precisely herein.