$\\mathbf{ab}$ -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 inthe ring $R$ is a noncommutative polynomial $\\Psi(P)$ inthe free associative algebra $R\\langle\\mathbf{a},\\mathbf{b}\\rangle$ defined by the formula

\\Psi(P)=\\sum_{c=\\{\\hat{0}=x_{0}<x_{1}<\\dots<x_{k}=\\hat{1}\\}}w(c),

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 theface 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, correspondingto edges. Further, each vertex is incident with exactly two edges.Let $c=\\{\\hat{0}=x_{0}<\\cdots<x_{k}=\\hat{1}\\}$ be a chain in $P_{n}$ . Thereare four possibilities.

1.
$c=\\{\\hat{0}<\\hat{1}\\}$ . This chain does not include any elementsof 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.
$c$ 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.
$c$ 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.
$c$ includes a vertex and an edge. Since each vertex is incident withexactly two edges, this can happen in $2n$ ways. The weight of such achain 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 noncommutativepolynomial 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. Thusthe $\\mathbf{c}\\mathbf{d}$ -index of the $n$ -gon is $\\mathbf{c}^{2}+(n-2)\\cdot\\mathbf{d}$ . Notevery graded poset has a $\\mathbf{c}\\mathbf{d}$ -index. However, every poset whicharisesas the face lattice of a convex polytope, or more generally, every gradedposet 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 writtenin terms of $\\mathbf{c}$ and $\\mathbf{d}$ is the boolean algebra $B_{2}$ witha 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 forpolytopes, 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.

𝐚𝐛-index of graded posets

References

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