Place these permutations of {1,2,3,4,5,6} in lexicographic...

Key Concepts

Permutations

A permutation is an ordered arrangement of a set of items. In combinatorics, finding all permutations of a set involves determining every possible sequence in which the elements can appear, taking into account that the order matters.

Lexicographic Order

Lexicographic ordering is a method of sorting sequences by comparing elements from left to right, much like words are ordered in a dictionary. When applied to permutations, each sequence is compared digit by digit, and the first differing element determines the order of the sequences.

Comparison Process

The comparison process in lexicographic order involves examining corresponding positions of two sequences until a difference is found. The sequence with the smaller element at the point of difference is considered to come first in the order, which is essential for systematically arranging permutations.

Transcript

00:01 So we'd like to put some permutations in lexicographic order.

00:03 This is actually relatively straightforward.

00:05 That lexicography is the art or the construction of dictionaries.

00:10 So we're putting them in the order that they would be in if they were in a dictionary, which is the same thing as their numerical order.

00:16 So we're going to index first by the first digit in here.

00:19 Looks like we're going to get these two start with one and neither of the other ones do.

00:24 None of the other ones do.

00:25 So we just need to sort these.

00:28 That's going to be one five before one six.

00:30 So one five six four two three comes before one six five four three two.

00:39 Let's go ahead and cross these out because we're done.

00:43 Now i'm going to start highlighting instead of underlining in green so it doesn't get too messy.

00:46 I'm going to look at all the ones that start with two.

00:52 That's all those.

00:53 They all start with two three so we're going to have to go next two three one and then in fact these are both two three one four.

01:00 So we go two three one through five six before two three one four six five two three one four five six two three one four six five.

01:13 We're done with those.

01:16 I'll go ahead and unhighlight them as well if i can do that carefully.

01:20 And our remaining permutation starting with two is two three four five six one...

Please give Ace some feedback