(y i 0 ,1 ) • • • (y i n ,n ) =count n (0, 0) =count n (0, 0)count n (-1, 0) + count n (0, 1)count n (-1, 1) bars of length 0 = ∞. Now, consider the case of finite length bars, i.e. for some 1 ≤ r ≤ m let us count the number of bars of length r < ∞. For a bar to have length r , two conditions must be satisfied. First, all building block bar lengths must be greater than or equal to r , i.e. we need i q ≤ r for all q. Second, at least one building block bar must come from the r th row of the matrix Y lengths , i.e., we require that i q = r for some q. This is because an intersection of (aligned) bars has length r if and only if every bar in the intersection has length at least r , with at least one bar having length exactly r . Via the Künneth formula, we have two types of finite length bars; standard bars and mathematical torsion bars. The number of standard bars of length r is the number of ways to combine any n bars of length at least r minus the number of ways to combine n bars of length strictly greater than r . The number of ways to do this is