The Weak Order on Integer Posets

The weak order on the symmetric group naturally extends to a lattice on all integer binary relations. We first show that the subposet of this weak order induced by integer posets defines as well a lattice. We then study the subposets of this weak order induced by specific families of integer posets corresponding to the elements, the intervals, and the faces of the permutahedron, the associahedron, and some recent generalizations of those. The weak order is the lattice on the symmetric group S(n) defined as the inclusion order of inversions, where an inversion of σ ∈ S(n) is a pair of values a < b such that σ−1(a) > σ−1(b). It is a fundamental tool for the study of the symmetric group, in connection to reduced expressions of permutations as products of simple transpositions. It can also be seen as an orientation of the skeleton of the permutahedron (the convex hull of all permutations of S(n) seen as vectors in R). The weak order naturally extends to all integer binary relations, i.e. binary relations on [n]. Namely, for any two integer binary relations R,S on [n], we define in this paper R 4 S ⇐⇒ R ⊇ S and R ⊆ S, where R := {(a, b) ∈ R | a ≤ b} and R := {(b, a) ∈ R | a ≤ b} respectively denote the increasing and decreasing subrelations of R. We call this order the weak order on integer binary relations, see Figure 1. The central result of this paper is the following statement, see Figure 5. Theorem 1. The weak order on the integer posets on [n] is a lattice. Our motivation for this result is that many relevant combinatorial objects can be interpreted by specific integer posets, and the subposets of the weak order induced by these specific integer posets often correspond to classical lattice structures on these combinatorial objects. To illustrate this, we study specific integer posets corresponding to the elements, to the intervals, and to the faces in the classical weak order, the Tamari and Cambrian lattices [MHPS12, Rea06], the boolean lattice, and other related lattices defined in [PP16]. By this systematic approach, we rediscover and shed light on lattice structures studied by G. Chatel and V. Pons on Tamari interval posets [CP15], by G. Chatel and V. Pilaud on Cambrian and Schröder-Cambrian trees [CP14], by D. Krob, M. Latapy, J.-C. Novelli, H.-D. Phan and S. Schwer on pseudo-permutations [KLN01], and by P. Palacios and M. Ronco [PR06] and J.-C. Novelli and J.-Y. Thibon [NT06] on plane trees. Part 1. The weak order on integer posets 1.1. The weak order on integer binary relations 1.1.1. Integer binary relations. Our main object of focus are binary relations on integers. An integer (binary) relation of size n is a binary relation on [n] := {1, . . . , n}, that is, a subset R of [n]. As usual, we write equivalently (u, v) ∈ R or uR v, and similarly, we write equivalently (u, v) / ∈ R or u 6R v. Recall that a relation R ∈ [n] is called: • reflexive if u R u for all u ∈ [n], • transitive if u R v and v R w implies u R w for all u, v, w ∈ [n], • symmetric if u R v implies v R u for all u, v ∈ [n], • antisymmetric if u R v and v R u implies u = v for all u, v ∈ [n]. From now on, we only consider reflexive relations. We denote by R(n) (resp. T (n), resp. S(n), resp. A(n)) the collection of all reflexive (resp. reflexive and transitive, resp. reflexive and symmetric, resp. reflexive and antisymmetric) integer relations of size n. We denote by C(n) the set of integer congruences of size n, that is, reflexive transitive symmetric integer relations, and by P(n) VPi was partially supported by the French ANR grant SC3A (15 CE40 0004 01).