rank-selected poset

Let $P$ be a graded poset of rank $n+1$ with rank function $\rho$. For any $S\subset \{0,1,\mathrm{\dots },n+1\}$ let $P_S$ denote the subset

$$P_S=\{x\in P:\rho (x)\in S\}={\rho }^{-1}(S).$$

Each such subset inherits a poset structure from $P$ as an induced poset. So we call $P_S$ 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 ${\alpha }_S(P)$ is the number of saturated chains in $P_S$. Then define ${\beta }_S(P)$ by

$${\beta }_S(P)=\sum _{T\subset S}{(-1)}^{|S|-|T|}{\alpha }_T(P).$$

The invariant $\beta$ 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 $\widehat{0}$ (representing the empty face) and $\widehat{1}$ (representing the interior of the polytope). For any $i\in \{0,\mathrm{\dots },n-1\}$, the alpha invariant ${\alpha }_{\{i+1\}}(L)$ counts the number of faces of $P$ of dimension $i$. For arbitrary $S\subset \{1,\mathrm{\dots },n\}$, the numbers ${\alpha }_S(L)$ are entries in the flag $f$-vector of $P$ and thus count flags of faces in $P$, while the ${\beta }_S(L)$ 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.