OR Why was the English army small enough that it could be defeated by a French duke?
Surely as duke of Normandy, William would have had access to much fewer men than the king of England. Did William hire mercenaries or bring in allies from elsewhere? Was England in some kind of state of weakness, other than having just fought off Harald Hardrada?
See my comment below for more details, but keep in mind that Normandy wasn't sparsely populated, had lots of money, and powerful allies. William also wasn't under the control of The Capetian monarch, Philip I - google Vexin for an example. Medieval titles do not indicate a rigid political hierarchy like you seem to infer. First, there were a lot to choose from, but the most popular were dux, rex, comte, marquis, ealdorman, and some others (I should note that later in the middle ages these things did become complicated and more structured). At this point, however, Dux Willelmus Normandi, as he might have been called in a chronicle literally means "William, leader of Normandy," which is what he was, and he ran his territory fairly autonomously, even if his ancestors had paid and owed some type of homage to the Capetian royal family.
In addition, after Edward the Confessor died earlier in 1066, the stability of the English kingdom went out the window as Harold, Tostig, and Harald fought for control, making it more difficult for Harold to gather troops - troops he was sure were loyal to him. So even if there were more English troops, Harold was in a poor political position in which to make the best use of them. William, on the other hand, had quelled the unrest in Normandy earlier in his reign, which allowed him organize the necessary troops and supplies to come to England, though his claim was a rather flimsy one - that Edward the Confessor said he could be king when he stayed with William during his exile.
Here are a couple of book that can give you more information since most of the comments seem to have been deleted:
George Garnett. The Norman Conquest: A Very Short Introduction
Peter Rex. 1066: A New History of the Norman Conquest (Popular, but good, history)
S. Morillo (ed.). The Battle of Hastings: Sources and Interpretations (Woodbridge, 1996)
D.C. Douglas. William the Conqueror (UC Berkeley, 1964) -- definitive and classic, but dated.
Christopher Gravett; Ivan Lapper. The Norman conquest (Oxford, Osprey, 2000) -- shows troops formations and other pretty pictures.
Hope this helps a little. Happy Reading!