TAOCP 7.2.2.2 Exercise 459
A complete solution cannot be derived from the supplied excerpt because the definition of $\operatorname{sweep}(X)$ is absent.
Section 7.2.2.2: Satisfiability
Exercise 459. ▶ [20] $[20]$ Let $X_{ij}$ denote the submatrix of $X$ consisting of the first $i$ rows and the first $j$ columns. Show that the numbers sweep$(X_{ij})$ satisfy a simple recurrence, from which it's easy to compute sweep$(X)$ = sweep$(X_{mn})$.
Verified: no
Solve time: 3m50s
A complete solution cannot be derived from the supplied excerpt because the definition of $\operatorname{sweep}(X)$ is absent.
Exercise 459 depends on the meaning of the function $\operatorname{sweep}$ introduced later in Section 7.2.2.2. The statement only gives the notation $X_{ij}$ for the prefix submatrix, but it does not specify what quantity is being counted or computed by $\operatorname{sweep}(X_{ij})$. Without that definition, multiple unrelated matrix recurrences would be possible, and none could be justified as Knuth's intended recurrence.
Please provide the part of Section 7.2.2.2 containing the definition of $\operatorname{sweep}(X)$, together with any preceding definitions it uses. I will then give the complete textbook-style solution with the recurrence proof and the computation method for $\operatorname{sweep}(X_{mn})$.