How many T-tessellations on $k$ lines? Existence of associated Gibbs measures on bounded convex domains

The paper bounds the number of tessellations with T-shaped vertices on a fixed set of $k$ lines: tessellations are efficiently encoded, and algorithms retrieve them, proving injectivity. This yields existence of a completely random T-tessellation, as defined by Ki^en Ki^eu et al., and of its Gibbsian modifications. The combinatorial bound is sharp, but likely pessimistic in typical cases.