TAOCP 5.1.2 Exercise 4
If $d < c < b < a$, the canonical factorization of (12) is obtained by reversing the order of the letters in each cycle of the factorization given in (17).
Section 5.1.2: Permutations of a Multiset
Exercise 4. [**] [M11] The canonical factorization of (12), in the sense of Theorem A, is given in (17) when $a < b < c < d$. Find the corresponding canonical factorization when $d < c < b < a$.
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Solve time: 18m56s
If $d < c < b < a$, the canonical factorization of (12) is obtained by reversing the order of the letters in each cycle of the factorization given in (17). Each cycle is written with its elements in increasing order according to the new total order $d < c < b < a$, so that the smallest element of the cycle appears first. Explicitly, every occurrence of $(a,b)$ becomes $(b,a)$, $(a,c)$ becomes $(c,a)$, $(a,b,c)$ becomes $(c,b,a)$, and so forth. The overall factorization preserves the relative positions of the cycles, because Theorem A depends only on disjointness and not on the global ordering of letters. This completes the answer. ∎