Finding a pattern an exact number of times and excluding patterns

Question

I have a list of expressions which are multiplications of different types of objects. The question is broadly about how I can look for a pattern an exact number of times and exclude patterns from the search.

I will start with an example list on which I am going to base my question:

lyst = {n1 * n2 * Pair[M[p1],M[s2]] * Pair[M[p3],M[s4]], n3 * n4 * Pair[M[p1],M[s2]] * Pair[M[p1],M[s2]], n5 * n6 * Pair[M[p1],M[s2]] }

Here, n1 - n4 are numbers and "Pair" and "M" are Heads that are defined in my program. Now what I would like to do is select all elements, in which the expression Pair[M[p1],M[s2]] appears exactly N times without objects whose head is 'Pair' appearing again (subsequently I also want to do this with other expressions but this is just to understand how I should tackle this problem).

Example: I would like to have a pattern test that looks for all elements in 'lyst' which contain Pair[M[p1],M[s2]] without the structure Pair[....] appearing again. In my example list that would just be the last element only, since it does contain this expression and 'Pair[...]' does not occur again.

I would be very thankful for suggestions on how to do this since I do not really have a good idea so far.

Answer 1

Note, if Pair[M[p1],M[s2]] appears n times, it will be simplified to Pair[M[p1],M[s2]]^n. Therefore, the pattern to search must contain Pair[M[p1],M[s2]]^n. Further we must account for additional factors by "___" Here is an example with multiple items:

ex = {Times[n1, n2, Pair[M[p1], M[s2]]], 
   Times[n1, Pair[M[p1], M[s2]], Pair[M[p1], M[s2]], 
    Pair[M[p1], M[s2]]], 
   Times[n3, n4, Z[1], Pair[M[p1], M[s2]], Pair[M[p1], M[s2]]]} 

{n1 n2 Pair[M[p1], M[s2]], n1 Pair[M[p1], M[s2]]^3, 
 n3 n4 Pair[M[p1], M[s2]]^2}

For n==1:

Cases[ex, ___ Pair[M[p1], M[s2]]]

{n1 n2 Pair[M[p1], M[s2]]}

For N==2:

Cases[ex, ___ Pair[M[p1], M[s2]]^2]

{n3 n4 Pair[M[p1], M[s2]]^2}

For n==3:

Cases[ex, ___ Pair[M[p1], M[s2]]^3]

{n1 Pair[M[p1], M[s2]]^3}

Addendum

If there are several different Pair[..] in the same element:

lyst = {n1*n2*Pair[M[p1], M[s2]]*Pair[M[p3], M[s4]], 
  n3*n4*Pair[M[p1], M[s2]]*Pair[M[p1], M[s2]], 
  n5*n6*Pair[M[p1], M[s2]]}

Cases[lyst, 
 HoldPattern[Except[___Pair] Pair[M[p1], M[s2]] Except[___Pair] ]]

{n5 n6 Pair[M[p1], M[s2]]}