How to remove reverse rows in a permutation matrix? - arrays

Question

I'm looking for a quick way in MATLAB to do the following:
Given a permutation matrix of a vector, say [1, 2, 3], I would like to remove all duplicate reverse rows.
So the matrix P = perms([1, 2, 3])
3 2 1
3 1 2
2 3 1
2 1 3
1 3 2
1 2 3
becomes
3 2 1
3 1 2
2 3 1

Answer 1

You can noticed that, symetrically, the first element of each rows have to be bigger than the last one:
n = 4; %row size
x = perms(1:n) %all perms
p = x(x(:,1)>x(:,n),:) %non symetrical perms
Or you can noticed that the number of rows contained by the p matrix follows this OEIS sequence for each n and correspond to size(x,1)/2 so since perms output the permutation in reverse lexicographic order:
n = 4; %row size
x = perms(1:n) %all perms
p = x(1:size(x,1)/2,:) %non symetrical perms

Answer 2

You can use MATLAB's fliplr method to flip your array left to right, and then use ismember to find rows of P in the flipped version. At last, iterate all locations and select already found rows.
Here's some code (tested with Octave 5.2.0 and MATLAB Online):
a = [1, 2, 3];
P = perms(a)
% Where can row x be found in the left right flipped version of row x?
[~, Locb] = ismember(P, fliplr(P), 'rows');
% Set up logical vector to store indices to take from P.
n = length(Locb);
idx = true(n, 1);
% Iterate all locations and set already found row to false.
for I = 1:n
if (idx(I))
idx(Locb(I)) = false;
end
end
% Generate result matrix.
P_star = P(idx, :)
Your example:
P =
3 2 1
3 1 2
2 3 1
2 1 3
1 3 2
1 2 3
P_star =
3 2 1
3 1 2
2 3 1
Added 4 to the example:
P =
4 3 2 1
4 3 1 2
4 2 3 1
4 2 1 3
4 1 3 2
4 1 2 3
3 4 2 1
3 4 1 2
3 2 4 1
3 2 1 4
3 1 4 2
3 1 2 4
2 4 3 1
2 4 1 3
2 3 4 1
2 3 1 4
2 1 4 3
2 1 3 4
1 4 3 2
1 4 2 3
1 3 4 2
1 3 2 4
1 2 4 3
1 2 3 4
P_star =
4 3 2 1
4 3 1 2
4 2 3 1
4 2 1 3
4 1 3 2
4 1 2 3
3 4 2 1
3 4 1 2
3 2 4 1
3 1 4 2
2 4 3 1
2 3 4 1
As demanded in your question (at least from my understanding), rows are taken from top to bottom.

Answer 3

Here's another approach:
result = P(all(~triu(~pdist2(P,P(:,end:-1:1)))),:);
pdist computes the distance between rows of P and rows of P(:,end:-1:1).
~ negates the result, so that true corresponds to coincident pairs.
triu keeps only the upper triangular part of the matrix, so that only one of the two rows of the coincident pair will be removed.
~ negates back, so that true corresponds to non-coincident pairs.
all gives a row vector with true for rows that should be kept (because they do not coincide with any previous row).
This is used as a logical index to select rows of P.