Between weak and Bruhat: the middle order on permutations

TL;DR

This paper introduces the 'middle order', a new partial order on permutations that refines both the weak and Bruhat orders, and explores its combinatorial properties, lattice structure, and special cases like involutions.

Contribution

It defines the middle order on permutations, shows it refines existing orders, and analyzes its lattice and combinatorial properties, including intervals and Euler characteristic.

Findings

Middle order is a distributive lattice.

Characterization and enumeration of intervals and boolean intervals.

Simple formula for the M"obius function on involutions.

Abstract

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 M\"obius 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 $\mathcal{P}_n$.