You are welcome and welcome to the forum.

]]>123: 2 ascents

132: 1 ascent

213: 1 ascent

231: 1 ascent

312: 1 ascent

321: 0 ascents

How would you write the answer? Just like this?

]]>2 >1 so that is an ascent, then 3 > 2 that is another ascent. So 1,2,3 has 2 ascents.

Now look at 3,2,1. 3 is not less than 2 and 2 is not less than 1. 3,2,1 has no ascents.

Then just want you to look at all 6 and find the ones with 0 ascents, 1 ascent and 2 ascents.

]]>Thank you for your quick reply!

]]>Did you try writing down all the permutations, there are only 6?

Can you do it now?

]]>Consider the permutation of 1, 2, 3, 4. The permutation 1432, for instance, is said to have one ascent namely, 14 (since 1 < 4). This same permutation also has two descents namely, 43 (since 4 > 3) and 32 (since 3 > 2). The permutation 1423, on the other hand, has two ascents, at 14 and 23 and the one descent 42.

a) How many permutations of 1, 2, 3 have k ascents, for k = 0, 1, 2?

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