In 2021 Fang and Reineke described the support of linear degenerations of flag varieties in terms of Motzkin paths, by using Knight-Zelevinsky multi-segment duality. In a joint project with Esposito and Marietti (IMRN 2023, arXiv 2206.10281) we give a new characterization of supports in representation-theoretic terms by what we call excessive multi-segments. To do so we consider an algebraic structure on the set of Motzkin paths that we call Motzkin monoid. By using a universal property of the Motzkin monoid, we show that excessive multi segments are parametrized in a natural way by Motzkin paths. Moreover, we show that this parametrization coincides exactly with the Fang-Reineke parametrization. As a byproduct we have an elementary combinatorial criterion to decide if a multisegment is a support. We have an inductive procedure to describe the inverse of the Fang-Reineke map. In this term there is a very beautiful (as yet conjectural) formula for the coefficients.