If I’m understanding the definition of a first run correctly, then the lengths of the first runs of the permutations of are as follows:

123: 3

132: 2

213: 2

231: 2

312: 1

321: 1

Thus, the average length of a first run is , but . Am I doing something wrong?

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I think the permutation 213 has 1 as the length of its leftmost run. The 2 by itself, yes? This should give for the average length, which does match up.

Isn’t 23 the leftmost run of 213? It doesn’t seem like the in the definition are required to be consecutive.

@dgklein But then how do you distinguish “leftmost” run from any other run? I think a leftmost run must begin at the beginning, right?

@dgklein Sorry, I think I misunderstood your point…now I’m confused too

I did it for and it seems they have to be consecutive…

Sorry guys, I wrote HW3 on an airplane, and apparently I wasn’t writing very lucidly. I already fixed the mistake on the website.

In problem 1(b), a run in a permutation is a maximal sequence of increasing **consecutive** elements . So the runs of 213 are 2 and 13.

On the other hand, I stated problem 1(d) slightly vaguely (what if there is no th run? Etc.). I did that on purpose. It's good for you to get used to slightly open-ended question. 🙂

So, for the example of 213, the leftmost run is 2?

Anastasia, that’s correct.