# isEulerian(Poset) -- determines if a ranked poset is Eulerian

## Synopsis

• Function: isEulerian
• Usage:
i = isEulerian P
• Inputs:
• P, an instance of the type Poset, a ranked poset
• Outputs:
• i, , whether $P$ is Eulerian

## Description

The poset $P$ is Eulerian if every non-trivial closedInterval of $P$ has an equal number of vertices of even and odd rank.

The $n$ chain is non-Eulerian for $n \geq 3$.

 i1 : isEulerian chain 10 o1 = false

The facePoset of the simplicialComplex of an $n$ cycle is Eulerian.

 i2 : n = 10; i3 : R = QQ[x_0..x_(n-1)]; i4 : F = facePoset simplicialComplex apply(n, i -> x_i * x_((i+1)%n)); i5 : isEulerian F o5 = true