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We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size
L
×
N
, with boundary conditions that are free in the
L-direction and periodic in the
N-direction. The decomposition of the partition function in terms of the characters
K
1
+
2
l
(with
l
=
0
,
1
,
…
,
L
) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the
K
1
+
2
l
thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly
j clusters are non-contractible with respect to the periodic lattice direction, and a partition function with fixed transverse boundary conditions.