---
abstract: 'We study some properties of the [**cd**]{}-index of the Boolean lattice. They are extremely similar to the properties of the [[[[**a**]{}]{}[[**b**]{}]{}]{}]{}-index, or equivalently, the flag $h$-vector of the Boolean lattice and hence may be viewed as their [**cd**]{}-analogues. We define a different algebra structure on the polynomial algebra $k \langle {{\bf c}}, {{\bf d}}\rangle$ and give a derivation on this algebra. It is of significance for the Boolean lattice and forms our main tool. Using similar methods, we also prove some results for the [**cd**]{}-index of the cubical lattice. We show that the Dehn-Sommerville relations for the flag $f$-vector of an Eulerian poset are equivalent to certain simple identities that exist in our algebra.'
address: |
    Department of Mathematics\
    Cornell University\
    Ithaca, NY 14853
author:
- Swapneel Mahajan
title: 'The [**cd**]{}-index of the Boolean lattice'
---

[^1]

Introduction
============

The [**cd**]{}-index is a non-commutative polynomial in the variables [**c**]{} and [**d**]{} which efficiently encodes the flag $f$-vector (equivalently the flag $h$-vector) of an Eulerian poset. The flag $h$-vector and the [**cd**]{}-index are mysterious objects with many interesting properties. It is true, for example, that the [**cd**]{}-index (and hence the [**ab**]{}-index) of the face lattice of a convex polytope 