# array – Finding number of possible arrangements of books on a shelf with a Monte-Carlo approach

The Problem

I am trying to determine numerically how many possible ways a number of books can be arranged on a shelf. Specifically, there are x3 categories “physics“, “sci-fi“, and “travel“. Each contains `N_phys`, `N_scifi`, and `N_travel` numbers of books respectively. Within their category, the books can be placed in any order, but they must stay within their respective category (i.e all the physics books together). The categories can then be arranged in any order (i.e I could have all the sci-fi books first, followed by travel, followed by physics, for example).

My Attempt

I have decided to label both each book and its category by integers, with “physics” = 1, “sci-fi” = 2, and “travel” = 3. The shelf is then a 2D array. So for example, a whole shelf could look like the following:

``````(3   2   1   4   1   2   2   1   3;   % Book ID
1   1   1   1   3   3   2   2   2)  % Categrory label
``````

where the the first 4 books are physics (because the second row is 1), followed by 2 travel books, and finally 3 sci-fi books, like this: This problem can be solved easily and exactly as a permutation, and the result is `N_phys ! x N_scifi ! x N_travel ! x 3 !`. For `N_phys = 4`, `N_scifi = 3`, and `N_travel = 2`, the result is 1728 possible arrangements.

I have written a “brute-force” numerical attempt, shown below in Matlab, which seems to give the correct result:

``````N_phys = 4;   % Number of physics books
N_scifi = 3;  % Number of sci-fi books
N_travel = 2; % Number of travel books

books_physics = (1:N_phys;...
ones(1,N_phys)); % Collection of physics books

books_scifi = (1:N_scifi;...
2*ones(1,N_scifi)); % Collection of sci-fi books

books_travel = (1:N_travel;...
3*ones(1,N_travel));  % Collection of travel books

num_samples = 10000; % Number of random shelves to generate
unique_shelves = zeros(num_samples*2,N_phys+N_scifi+N_travel); % Preallocate

unique_shelves(1:2,:) = (books_physics books_scifi books_travel);
num_unique_shelves = 1;

% Generate "num_samples" permutations of shelves randomly
for sample_num = 1:num_samples

books_physics_shuffled = books_physics( :, randperm(N_phys) ); % Shuffle physics books
books_scifi_shuffled = books_scifi( :, randperm(N_scifi) );    % Shuffle sci-fi books
books_travel_shuffled = books_travel( :, randperm(N_travel) ); % Shuffle travel books

category_order = randperm(3,3); % Choose order of categories, e.g. sci-fi/phsycis/travel  = (2 1 3)

shelf = ();

% Arrange the categories in a random order
for k = 1:3
if category_order(k) == 1
shelf = (shelf books_physics_shuffled);
elseif category_order(k) == 2
shelf = (shelf books_scifi_shuffled);
elseif category_order(k) == 3
shelf = (shelf books_travel_shuffled);
end
end

% Iterate over discovered shelves, and see if we have found a new unique one
shelf_exists = 0;
for k = 1:num_unique_shelves
if shelf == unique_shelves( (2*k-1):(2*k),:)
shelf_exists = 1; % Shelf was previously discovered
break
end
end

if ~shelf_exists % New shelf was found
unique_shelves( (2*num_unique_shelves+1):(2*num_unique_shelves+2),:) = shelf; % Add shelf to existing ones
num_unique_shelves = num_unique_shelves + 1;
end

end

disp(('Number of unique shelves found = ',num2str(num_unique_shelves)))
disp(('Expected = ', num2str(factorial(N_phys)*factorial(N_scifi)*factorial(N_travel)*factorial(3) )))
``````

As can be seen, I am basically randomly generating a shelf, and then checking if it has been previously found. If it has, I add it to the list.

I am looking for feedback on the way this is implemented, and how it can be improved to make it more concise and efficient. Is there a better data structure for storing such “unique_shelves”, instead of the 2D array of integers as above? My code also doesn’t scale easily for more categories, since they are hardcoded.

Tips or alternative examples would be great! Thanks.

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