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].

Should this instead read as follows:

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?

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4 comments

  1. I do not think it is a typo.

    • 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}.

  2. I was wondering about the same thing…We don’t usually put commas in the one line notation, do we? Like in the example in the footnote.

  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).

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