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The partitions can again be coded by means of a sequence of integers 1 , 2 $^{,}$· · · , d S $_{L}$. The coding algorithm is the same as that used in the consideration of the Potts model with vacancies [18] and in the study of site percolation [24]. 

To determine the critical threshold of the Potts model and/or site percolation, we calcu- late the magnetic scaled gap 

where ξ$_{h}$ ( v, L ) is the magnetic correlation length (with v replaced by s for site percolation). In the language of the random cluster model and site percolation, the magnetic correlation function is defined to be the probability that two sites at a distance r belong to the same cluster, or ′ 

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where Z ′ = ∑ ′ g v n b ( g $^{)}$q n c ( g ) for the random cluster model and Z ′ = ∑ ′ g s n s ( g $^{)}$(1 − s ) N − n s ( g ) for site percolation. The summations in Z ′ are the same as in (9) and (15) but restricted to subgraphs g with at least one cluster spanning from row 1 to row r . 

We define a transfer matrix, hereafter referred to as the magnetic sector of the transfer matrix, based on Z $^{′}$, in a way similar to that of the transfer matrix based on Z in (13) or (15) in the ’non-magnetic’ sector. In constructing the magnetic sector of the transfer matrix, we use the ‘magnetic’ type connectivity of the L end sites of the cylinder, which, in addition to describing how sites are connected, specifies which sites are still connected to a site in row 1. These sites are called ‘magnetic sites’. To count the total number of non- crossing partitions describing the ‘magnetic’ type connectivity, we first code the positions of the magnetic sites by means of a binary number m = 0 , 1 , ..., 2 $^{L}$− 1, where the binary digit 1 denotes a magnetic site. The magnetic sites divide the remaining sites in g ( m ) groups such that two sites in different groups cannot be connected. Let n ( j ) be the number of sites in the j -th group. Then there are 

non-crossing partitions for the Potts model and 

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non-crossing partitions for site percolation. The total number of non-crossing partitions is 

