# Hw3 Problem number 5b typo?

For problem number 5b it reads:

The product of the permutations $a_{1}, \dots, a_n$ of $[n]$ and $b_1, \dots, b_m$ of $[m]$ (in one-line notation) is the permutation  $a_1, \dots, a_n, b_1+n, \dots, b_m+n$ of $[n+m]$.

The product of the permutations $a_{1}, \dots, a_n$ of $[n]$ and $b_1, \dots, b_m$ of $[m]$ (in one-line notation) is the permutation $a_1, \dots, a_n, b_{1+n}, \dots, b_{m+n}$ of $[n+m]$.

So should the addition of the $n$ on the $b$ elements be a subscript?

• Consider the following example: Take the sets $A = [2]$ and $B = [5]$ and consider the permutations $\pi = 2, 1$ and $\rho = 2, 3, 4, 1, 5$ –both in one-line notation.
Then the product of the above two permutations would be $2, 1, 4, 5, 6, 3, 7$, which is indeed a permutation of $[2 + 5] = [7].$ If we take your suggestion, I don’t see how the multiplication would make sense because we don’t have $b_{2 + 5} = b_{7}$.
3. The problem is correct as stated. As for commas in one-line notation, they are necessary if the terms are more complicated than just one-digit numbers or single variables, as in this case. Without commas, the permutation would be an unreadable mess: $\left(a_1...a_nb_1+1...+b_m+n\right)$.