Combinatorics of Permutreehedra and Geometry of $s$-Permutahedra

Abstract: This thesis finds its place in the interplay between algebraic and geometric combinatorics. We focus on studying two different families of lattices in relation to the weak order: the permutree lattices and the $s$-weak order. The first part involves the permutree quotients of the weak order. We define inversion and cubic vectors on permutrees which respectively give a constructive meet operation between permutrees and a cubical realization of permutreehedra. We characterize minimal elements of permutree congruence classes using automata that capture ${ijk}/{kij}$-pattern avoidances and generalize stack sorting and Coxeter sorting. The second part centers on flow polytopes. More specifically, we give a positive answer to a conjecture of Ceballos and Pons on the $s$-permutahedron when $s$ is a composition. We define the $s$-oruga graph whose flow polytope recovers the $s$-weak order with explicit coordinates. Finally, we introduce the bicho graphs whose flow polytopes describe permutree lattices.