-index of graded posets
Let be a graded poset of rank with a and a . Let be the rank function of . The -index of with coefficients in the ring is a noncommutative polynomial in the free associative algebra defined by the formula
with the weight of a chain defined by , where
Let us compute in a simple example. Let be the face lattice of an -gon. Below we display .
Thus has atoms, corresponding to vertices, and coatoms, corresponding to edges. Further, each vertex is incident with exactly two edges. Let be a chain in . There are four possibilities.
. This chain does not include any elements of ranks 1 or 2, so its weight is .
includes a vertex but not an edge. This can happen in ways. Each such chain has weight .
includes an edge but not a vertex. This can also happen in ways. Each such chain has weight .
includes a vertex and an edge. Since each vertex is incident with exactly two edges, this can happen in ways. The weight of such a chain is .
Summing over all the chains yields
In this case the -index can be rewritten as a noncommutative polynomial in the variables and . When this happens, we say that has a -index. Thus the -index of the -gon is . Not every graded poset has a -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 -index.
The -index of this poset is .
- 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 -vectors and the -index, Math. Z. 216 (1994), 483-499.
|Title||-index of graded posets|
|Date of creation||2013-03-22 15:46:47|
|Last modified on||2013-03-22 15:46:47|
|Last modified by||mps (409)|