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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, a Boolean value, 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

See also

closedInterval -- computes the subposet contained between two points
isRanked -- determines if a poset is ranked
moebiusFunction -- computes the Moebius function at every pair of elements of a poset

Ways to use this method:

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