# $\mathbf{ab}$-index of graded posets

Let $P$ be a graded poset of rank $n+1$ with a $\hat{0}$ and a $\hat{1}$. Let $\rho\colon P\to\mathbb{N}$ be the rank function of $P$. The $\mathbf{ab}$-index of $P$ with coefficients in the ring $R$ is a noncommutative polynomial $\Psi(P)$ in the free associative algebra $R\langle\mathbf{a},\mathbf{b}\rangle$ defined by the formula

 $\Psi(P)=\sum_{c=\{\hat{0}=x_{0}

with the weight of a chain $c$ defined by $w(c)=z_{1}\cdots z_{n}$, where

 $z_{i}=\begin{cases}\mathbf{b},&i\in\rho(x_{0},\dots,x_{k})\\ \mathbf{a}-\mathbf{b},&\text{otherwise}.\end{cases}$

Let us compute $\Psi$ in a simple example. Let $P_{n}$ be the face lattice of an $n$-gon. Below we display $P_{5}$.

 $\xymatrix{&&\hat{1}\ar@{-}[lld]\ar@{-}[ld]\ar@{-}[d]\ar@{-}[rd]\ar@{-}[rrd]&&% \\ \{p,q\}\ar@{-}[d]\ar@{-}[rrrrd]&\{q,r\}\ar@{-}[ld]\ar@{-}[d]&\{r,s\}\ar@{-}[ld% ]\ar@{-}[d]&\{s,t\}\ar@{-}[ld]\ar@{-}[d]&\{t,u\}\ar@{-}[ld]\ar@{-}[d]\\ \{p\}\ar@{-}[rrd]&\{q\}\ar@{-}[rd]&\{r\}\ar@{-}[d]&\{s\}\ar@{-}[ld]&\{t\}\ar@{% -}[lld]\\ &&\hat{0}&&}$

Thus $P_{n}$ has $n$ atoms, corresponding to vertices, and $n$ coatoms, corresponding to edges. Further, each vertex is incident with exactly two edges. Let $c=\{\hat{0}=x_{0}<\cdots be a chain in $P_{n}$. There are four possibilities.

1. 1.

$c=\{\hat{0}<\hat{1}\}$. This chain does not include any elements of ranks 1 or 2, so its weight is $(\mathbf{a}-\mathbf{b})^{2}=\mathbf{a}^{2}-\mathbf{a}\mathbf{b}-\mathbf{b}% \mathbf{a}+\mathbf{b}^{2}$.

2. 2.

$c$ includes a vertex but not an edge. This can happen in $n$ ways. Each such chain has weight $\mathbf{b}(\mathbf{a}-\mathbf{b})$.

3. 3.

$c$ includes an edge but not a vertex. This can also happen in $n$ ways. Each such chain has weight $(\mathbf{a}-\mathbf{b})\mathbf{b}$.

4. 4.

$c$ includes a vertex and an edge. Since each vertex is incident with exactly two edges, this can happen in $2n$ ways. The weight of such a chain is $b^{2}$.

Summing over all the chains yields

 $\displaystyle\Psi(P)$ $\displaystyle=\mathbf{a}^{2}+(n-1)\cdot\mathbf{a}\mathbf{b}+(n-1)\cdot\mathbf{% b}\mathbf{a}+\mathbf{b}^{2}$ $\displaystyle=(\mathbf{a}+\mathbf{b})^{2}+(n-2)\cdot(\mathbf{a}\mathbf{b}+% \mathbf{b}\mathbf{a}).$

In this case the $\mathbf{a}\mathbf{b}$-index can be rewritten as a noncommutative polynomial in the variables $\mathbf{c}=\mathbf{a}+\mathbf{b}$ and $\mathbf{d}=\mathbf{a}\mathbf{b}+\mathbf{b}\mathbf{a}$. When this happens, we say that $P$ has a $\mathbf{c}\mathbf{d}$-index. Thus the $\mathbf{c}\mathbf{d}$-index of the $n$-gon is $\mathbf{c}^{2}+(n-2)\cdot\mathbf{d}$. Not every graded poset has a $\mathbf{c}\mathbf{d}$-index. However, every poset which arises as the face lattice of a convex polytope, or more generally, every graded poset which satisfies the generalized Dehn-Sommerville relations, has a $\mathbf{c}\mathbf{d}$-index.

An example of a poset whose $\mathbf{a}\mathbf{b}$-index cannot be written in terms of $\mathbf{c}$ and $\mathbf{d}$ is the boolean algebra $B_{2}$ with a new maximal element adjoined:

 $\xymatrix{&\hat{1}\ar@{-}[d]&\\ &\{0,1\}\ar@{-}[ld]\ar@{-}[rd]&\\ \{0\}\ar@{-}[rd]&&\{1\}\ar@{-}[ld]\\ &\hat{0}&}$

The $\mathbf{a}\mathbf{b}$-index of this poset is $\mathbf{a}^{2}+\mathbf{b}\mathbf{a}$.

## References

• 1 Bayer, M. and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets, Invent. Math. 79 (1985), no. 1, 143–157.
• 2 Bayer, M. and A. Klapper, A new index for polytopes, Discrete Comput. Geom. 6 (1991), no. 1, 33–47.
• 3 Stanley, R., Flag $f$-vectors and the $\mathbf{cd}$-index, Math. Z. 216 (1994), 483-499.
Title $\mathbf{ab}$-index of graded posets mathbfabindexOfGradedPosets 2013-03-22 15:46:47 2013-03-22 15:46:47 mps (409) mps (409) 6 mps (409) Topic msc 06A07 ab-index cd-index $\mathbf{ab}$-index $\mathbf{cd}$-index