-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.
-
1.
. This chain does not include any elements of ranks 1 or 2, so its weight is .
-
2.
includes a vertex but not an edge. This can happen in ways. Each such chain has weight .
-
3.
includes an edge but not a vertex. This can also happen in ways. Each such chain has weight .
-
4.
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.
An example of a poset whose -index cannot be written
in terms of and is the boolean algebra with
a new maximal element
adjoined:
The -index of this poset is .
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 -vectors and the -index, Math. Z. 216 (1994), 483-499.
Title | -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 | -index |
Synonym | -index |