rank-selected poset

Let P be a graded poset of rank n+1 with rank function ρ. For any S{0,1,,n+1} let PS denote the subset


Each such subset inherits a poset structureMathworldPlanetmath from P as an induced poset. So we call PS the rank-selected poset of P induced by S, or more briefly the S-rank-selected subposet of P.

The rank-selected posets of a poset P can be used to define two special arithmetic invariants of P. First for each S, the alpha invariant αS(P) is the number of saturated chains in PS. Then define βS(P) by


The invariant β is called the rank-selected Möbius invariant of P.

For example, let L be the face poset of a convex polytope P of dimension n, including the special elements 0^ (representing the empty face) and 1^ (representing the interior of the polytope). For any i{0,,n-1}, the alpha invariant α{i+1}(L) counts the number of faces of P of dimension i. For arbitrary S{1,,n}, the numbers αS(L) are entries in the flag f-vector of P and thus count flags of faces in P, while the βS(L) are entries in the flag h-vector of P.

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.


Title rank-selected poset
Canonical name RankselectedPoset
Date of creation 2013-03-22 16:23:43
Last modified on 2013-03-22 16:23:43
Owner mps (409)
Last modified by mps (409)
Numerical id 5
Author mps (409)
Entry type Definition
Classification msc 06A11
Classification msc 06A06
Defines alpha invariant
Defines beta invariant
Defines rank-selected Möbius invariant
Defines rank-selected Mobius invariant