Eulerian poset

Eulerian poset

A

graded poset

P

is

Eulerian

if for any

x

≤

y

in

P

, the
Möbius

function

(

http://planetmath.org/MoebiusInversionFormula

)
of the interval

[

x

,

y

]

is given by

μ

P

⁢

(

x

,

y

)

=

(

-

1

)

ρ

⁢

(

y

)

-

ρ

⁢

(

x

)

,

where

ρ

is the rank function of

P

. If

P

is finite and
graded with strictly

positive

rank, then we can equivalently and more simply say that

P

is
Eulerian if and only if each nontrivial interval of

P

has the same number of
elements of even rank as odd rank.

For example, the

lattice

(

http://planetmath.org/Lattice

)

M

3

is not Eulerian, because it
has 3 elements of rank 1 but only 2 elements of ranks 0 or 2.

\xymatrix

⁢

&

⁢

1

^

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

0

^

⁢

&

On the other hand, the

Boolean lattice

B

n

is always Eulerian.

\xymatrix

⁢

&

⁢

1

^

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

a

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

a

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

b

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

a

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

0

^

⁢

&

Each interval of

B

n

is

isomorphic

to a smaller Boolean lattice, so
we can give a proof by induction that

B

n

is Eulerian. Now

B

0

is
a single point and so there is only one interval, which has Möbius
function 1. So to show that

B

n

is Eulerian, we just need to show
that the Möbius function of

B

n

is

(

-

1

)

n

. Recall that by definition,

μ

B

n

⁢

(

B

n

)

=

-

∑

S

⊂

{

1

,

…

,

n

}

μ

⁢

(

∅

,

S

)

(*)

Thus

∑

S

⊆

{

1

,

…

,

n

}

μ

B

n

⁢

(

∅

,

S

)

=

0

. We can apply
the

binomial theorem

to

0

to see that

0

=

(

-

1

+

1

)

n

=

∑

k

=

0

n

(

n

k

)

⁢

(

-

1

)

k

and thus

(

-

1

)

n

=

-

∑

k

=

0

n

-

1

(

n

k

)

⁢

(

-

1

)

k

=

-

∑

S

⊂

{

1

,

…

,

n

}

μ

⁢

(

∅

,

S

)

(**)

It then follows from

Equation

(

*

)

and Equation

(

*

*

)

that

μ

⁢

(

B

n

)

=

(

-

1

)

n

.

Eulerian posets

are

generalizations

of certain special posets, such as
face lattices of polytopes.

Eulerian posets of small rank

In this

section

we classify

bounded

Eulerian posets of rank less than 4.

For

n

<

2

, there is exactly one

bounded poset

of rank

n

, the chain,
and it is Eulerian.

A bounded poset of rank

2

is determined precisely by its number of
coatoms. The

ab-index

of a rank

2

bounded Eulerian poset with

r

coatoms is

a

+

(

r

-

1

)

⁢

b

, which is a cd-index if and only if

r

=

2

.
Since every Eulerian poset has a cd-index, this implies that every
rank

2

bounded Eulerian poset has exactly two coatoms. Thus up to

isomorphism

B

2

is the only rank

2

bounded Eulerian poset.

Let

P

be a rank

3

bounded Eulerian poset. Since

P

has as many
elements of odd rank as even rank, it has exactly the same number of
atoms as coatoms. Since every rank

2

interval in

P

is a copy of

B

2

, it follows that

P

is not a chain and so has

n

≥

2

atoms.
Moreover, every atom is covered by exactly two coatoms, and every
coatom covers exactly two atoms. That is, the

Hasse diagram

of the
interior of

P

is a 2-regular

bipartite graph

. This graph admits a
decomposition into even cycles. In each cycle, half of the nodes are
atoms and half are coatoms. The non-induced subposet of

P

consisting of the points on a cycle of length

2

⁢

k

plus the minimum
and maximum elements of

P

is the face poset of a

k

-gon. Hence

P

itself is the face poset of the

disjoint union

of one or more

k

-gons
for various

k

≥

2

.

References

1

Stanley, R.,

Enumerative Combinatorics

, vol. 1, 2nd ed., Cambridge
University Press, Cambridge, 1996.

Title

Eulerian poset

Canonical name

EulerianPoset

Date of creation

2013-03-22 14:08:16

Last modified on

2013-03-22 14:08:16

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 06A11

Classification

msc 06A07

Related topic

GradedPoset

Generated on Fri Feb 9 09:53:33 2018 by

LaTeXML