Chapter 4, Problem 1E

To determine

The permutation of $\left\{1,2,3,4,5\right\}$ which follows 31524 and comes before 31524.

Answer to Problem 1E

The permutation 31524 is followed by $\underset{\_}{35124}$ and preceded by $\underset{\_}{31254}$.

Explanation of Solution

Algorithm used:

Begin with $\overleftarrow{1},\overleftarrow{2},\cdots ,\overleftarrow{n}$.

While there exists a mobile integer, do the following:

(1) Find the largest mobile integer m.

(2) Switch m and the adjacent integer to which its arrow points.

(3) Switch the direction of all the arrows above integers p with $p>m$.

Calculation:

The permutations of $\left(1,2,3,4\right)$ are listed in 24 rows.

According to the algorithm mentioned above, for each row $\left(1,2,3,\mathrm{.........24}\right)$, insert 5 in the five possible locations of the given permutation, working from right to left for odd numbered rows and left to right for even numbered rows.

The permutation 3124 appears on row 9 which is odd and so, for this row the 5's are inserted from right to left.

Thus, the permutation 31524 is followed by $\underset{\_}{35124}$ and preceded by $\underset{\_}{31254}$.