Efficient search for permutations that contain sub-permutations via array operations?

Question

I have a set of integers, say S = {1,...,10}, and two matrices N and M, whose rows are some (but not necessarily all possible) permutations of elements from S of orders, say

Answer 1

How about this?

n = size(N,1);
m = size(M,1);
p = size(N,2);
pattern = (1:p).'; %'// will be used for checking if it's a subpermutation or not
result = false(m,n); %// preallocate result, and initiallize to 0
for k = 1:size(N,1) %// loop over columns of (transposed) N
    [~, loc] = ismember(M, N(k,:)); %// entries of M equal to a value of N(:,k)
    ind = find(sum(loc>0,2)==p); %// we need p matches per row of M
    t = reshape(nonzeros(loc(ind,:).'),p,[]); %'// for those rows, take matches
    ind2 = all(bsxfun(@eq,t,pattern)); %// ... see if they are in the right order
    result(ind(ind2),k) = true; %// ... and if so, take note in result matrix
end

The result matrix contains 1 at position r,s if the s-th row N is a sub-permutation of the r-th row of M. From this, your desired results are

result1 = any(result,2);
result2 = sum(result,1);

Example:

M =
     8     9     4     1    10
     6     5     2     7     8
     4     1     9     2    10
     3     4     5     1     2

N =
     4     1     2
     4     9    10
     3     5     9

give

result =
     0     0     0
     0     0     0
     1     1     0
     1     0     0

result1 =
     0
     0
     1
     1

result2 =
     2     1     0