𝐚𝐛-index of graded posets

Let P be a graded poset of rank n+1 with a 0^ and a 1^. Let ρ:P be the rank function of P. The 𝐚𝐛-index of P with coefficientsMathworldPlanetmath in the ring R is a noncommutative polynomialMathworldPlanetmathPlanetmath Ψ(P) in the free associative algebra R𝐚,𝐛 defined by the formulaMathworldPlanetmathPlanetmath


with the weight of a chain c defined by w(c)=z1zn, where


Let us compute Ψ in a simple example. Let Pn be the face latticeMathworldPlanetmath of an n-gon. Below we display P5.


Thus Pn has n atoms, corresponding to vertices, and n coatoms, corresponding to edges. Further, each vertex is incident with exactly two edges. Let c={0^=x0<<xk=1^} be a chain in Pn. There are four possibilities.

  1. 1.

    c={0^<1^}. This chain does not include any elements of ranks 1 or 2, so its weight is (𝐚-𝐛)2=𝐚2-𝐚𝐛-𝐛𝐚+𝐛2.

  2. 2.

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

  3. 3.

    c includes an edge but not a vertex. This can also happen in n ways. Each such chain has weight (𝐚-𝐛)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 b2.

Summing over all the chains yields

Ψ(P) =𝐚2+(n-1)𝐚𝐛+(n-1)𝐛𝐚+𝐛2

In this case the 𝐚𝐛-index can be rewritten as a noncommutative polynomial in the variables 𝐜=𝐚+𝐛 and 𝐝=𝐚𝐛+𝐛𝐚. When this happens, we say that P has a 𝐜𝐝-index. Thus the 𝐜𝐝-index of the n-gon is 𝐜2+(n-2)𝐝. Not every graded poset has a 𝐜𝐝-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 relationsMathworldPlanetmath, has a 𝐜𝐝-index.

An example of a poset whose 𝐚𝐛-index cannot be written in terms of 𝐜 and 𝐝 is the boolean algebraMathworldPlanetmath B2 with a new maximal elementMathworldPlanetmath adjoined:


The 𝐚𝐛-index of this poset is 𝐚2+𝐛𝐚.


  • 1 Bayer, M. and L. Billera, Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered setsMathworldPlanetmath, 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 cd-index, Math. Z. 216 (1994), 483-499.
Title 𝐚𝐛-index of graded posets
Canonical name mathbfabindexOfGradedPosets
Date of creation 2013-03-22 15:46:47
Last modified on 2013-03-22 15:46:47
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Topic
Classification msc 06A07
Synonym ab-index
Synonym cd-index
Synonym 𝐚𝐛-index
Synonym 𝐜𝐝-index