Order, lattices, ordered algebraic structures Definition

Eulerian poset

Builds on Graded poset Bounded poset Möbius function of a poset

Definition

Let $P$ be a finite graded poset with rank function $\rho$ . Assume that $P$ is bounded, with a least element $\hat 0$ and a greatest element $\hat 1$ . For $x \le y$ in $P$ , let $[x, y] = \set{z \in P : x \le z \le y}$ . The poset $P$ is Eulerian when for all $x < y$ in $P$ , the interval $[x, y]$ has as many elements of even rank as of odd rank.

Remarks

Equivalently, the Möbius function of $P$ is $\mu(x, y) = (-1)^{\rho(y) - \rho(x)}$ for all $x \le y$ .

Example: the lattice of subsets of a finite set $S$ , with $\rho(A) = \abs{A}$ . Check: for $A \subsetneq B$ with $\abs{B \setminus A} = k \ge 1$ , the interval $[A, B]$ has $\sum_{j \text{ even}} \binom{k}{j} = 2^{k-1}$ elements of each parity of rank.

Non-example: the chain $0 < 1 < 2$ with $\rho(i) = i$ is graded and bounded, but the interval $[0, 2]$ has two elements of even rank and one of odd rank.

Every interval of an Eulerian poset is again Eulerian. The face lattice of a convex polytope is Eulerian.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`74JREVZVY3`
URI	`m2://74JREVZVY3`
Kind	definition
Writing	W0 stub
MSC	06A07, 52B05
Defines	Eulerian poset
Uses	Graded poset, Bounded poset, Möbius function of a poset

Eulerian poset

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here