Let be a graded poset of rank with rank function . For any let denote the subset
The invariant is called the rank-selected Möbius invariant of .
For example, let be the face poset of a convex polytope of dimension , including the special elements (representing the empty face) and (representing the interior of the polytope). For any , the alpha invariant counts the number of faces of of dimension . For arbitrary , the numbers are entries in the flag -vector of and thus count flags of faces in , while the are entries in the flag -vector of .
While the alpha invariant is by construction always nonnegative, the Möbius invariant is not guaranteed to be nonnegative. Posets for which the Möbius invariant is always nonnegative (and therefore counts something) are of special interest to combinatorialists. In particular, the Möbius invariant is nonnegative for face posets of convex polytopes.
- 1 Stanley, R., Enumerative Combinatorics, vol. 1, 2nd ed., Cambridge University Press, Cambridge, 1996.
|Date of creation||2013-03-22 16:23:43|
|Last modified on||2013-03-22 16:23:43|
|Last modified by||mps (409)|
|Defines||rank-selected Möbius invariant|
|Defines||rank-selected Mobius invariant|