Dynamic Programming on Secondary Structures concerned the minimum free energy of the RNA structure. Naturally, this extends to the partition function. (the standard notation for Partition Function in RNA secondary structure) is known to have the following relationship with free energy:
Using the summation principle, we have the following for , which is known as the corresponding secondary structure function for the subsequence enclosed by and with paired. (alternatively, the corresponding partition function for ).
where denotes that is a -cycle and corresponds to the pairs accessible by . The initial condition is that = 0.
The full partition function over with and not necessarily paired (the corresponding partition function for ) is given by the recursion
with the initial conditions = 1.0 and (i.e. the partition function of the empty subsequence is ).
However, it should be noted that these recursions still remain intractable and scales exponentially with .