Between Weak and Bruhat: The Middle Order on Permutations

We define a partial order $\mathcal{P}_n$ on permutations of any given size $n$, which is the image of a natural partial order on inversion sequences. We call this the “middle order.” We demonstrate that the poset $\mathcal{P}_n$ refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the Möbius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with $\mathcal{P}_n$, and defining a parking function analogue of $\mathcal{P}_n$.