𝐚𝐛-index of graded posets

𝐚𝐛

-index of graded posets

Let

P

be a

graded poset

of rank

n

+

1

with a

0

^

and a

1

^

.
Let

ρ

:

P

→

ℕ

be the rank function of

P

.
The

𝐚𝐛

-index

of

P

with

coefficients

in
the ring

R

is a noncommutative

polynomial

Ψ

⁢

(

P

)

in
the

free associative algebra

R

⁢

⟨

𝐚

,

𝐛

⟩

defined by the

formula

Ψ

⁢

(

P

)

=

∑

c

=

{

0

^

=

x

0

<

x

1

<

…

<

x

k

=

1

^

}

w

⁢

(

c

)

,

with the weight of a chain

c

defined by

w

⁢

(

c

)

=

z

1

⁢

⋯

⁢

z

n

, where

z

i

=

{

𝐛

,

i

∈

ρ

⁢

(

x

0

,

…

,

x

k

)

𝐚

-

𝐛

,

otherwise

.

Let us compute

Ψ

in a simple example. Let

P

n

be the
face

lattice

of an

n

-gon. Below we display

P

5

.

\xymatrix

⁢

&

⁢

&

⁢

1

^

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

]

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

]

⁢

&

⁢

&

⁢

{

p

,

q

}

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

r

⁢

r

⁢

d

]

⁢

&

⁢

{

q

,

r

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

{

r

,

s

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

{

s

,

t

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

{

t

,

u

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

{

p

}

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

]

⁢

&

⁢

{

q

}

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

{

r

}

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

{

s

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

{

t

}

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

]

⁢

&

⁢

&

⁢

0

^

⁢

&

⁢

&

Thus

P

n

has

n

atoms, corresponding to vertices, and

n

coatoms, corresponding
to edges. Further, each vertex is

incident

with exactly two edges.
Let

c

=

{

0

^

=

x

0

<

⋯

<

x

k

=

1

^

}

be a chain in

P

n

. There
are four possibilities.

1.

c

=

{

0

^

<

1

^

}

. This chain does not include any elements
of ranks 1 or 2, so its weight is

(

𝐚

-

𝐛

)

2

=

𝐚

2

-

𝐚𝐛

-

𝐛𝐚

+

𝐛

2

.

2.

c

includes a vertex but not an edge. This can happen in

n

ways.
Each such chain has weight

𝐛

⁢

(

𝐚

-

𝐛

)

.

3.

c

includes an edge but not a vertex. This can also happen in

n

ways.
Each such chain has weight

(

𝐚

-

𝐛

)

⁢

𝐛

.

4.

c

includes a vertex and an edge. Since each vertex is incident with
exactly two edges, this can happen in

2

⁢

n

ways. The weight of such a
chain is

b

2

.

Summing over all the chains yields

Ψ

⁢

(

P

)

=

𝐚

2

+

(

n

-

1

)

⋅

𝐚𝐛

+

(

n

-

1

)

⋅

𝐛𝐚

+

𝐛

2

=

(

𝐚

+

𝐛

)

2

+

(

n

-

2

)

⋅

(

𝐚𝐛

+

𝐛𝐚

)

.

In this case the

𝐚𝐛

-index can be rewritten as a noncommutative
polynomial in the

variables

𝐜

=

𝐚

+

𝐛

and

𝐝

=

𝐚𝐛

+

𝐛𝐚

.
When this happens, we say that

P

has a

𝐜𝐝

-index

. Thus
the

𝐜𝐝

-index of the

n

-gon is

𝐜

2

+

(

n

-

2

)

⋅

𝐝

. Not
every graded poset has a

𝐜𝐝

-index. However, every poset which
arises
as the face lattice of a convex polytope, or more generally, every graded
poset which

satisfies

the generalized Dehn-Sommerville

relations

, has a

𝐜𝐝

-index.

An example of a poset whose

𝐚𝐛

-index cannot be written
in terms of

𝐜

and

𝐝

is the

boolean algebra

B

2

with
a new

maximal element

adjoined:

\xymatrix

⁢

&

⁢

1

^

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

{

0

,

1

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

{

0

}

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

{

1

}

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

^

⁢

&

The

𝐚𝐛

-index of this poset is

𝐚

2

+

𝐛𝐚

.

References

1

Bayer, M. and L. Billera,

Generalized Dehn-Sommerville relations for
polytopes, spheres and Eulerian

partially ordered sets

, Invent. Math. 79
(1985), no. 1, 143–157.

2

Bayer, M. and A. Klapper,

A new index for polytopes

, Discrete Comput.
Geom. 6
(1991), no. 1, 33–47.

3

Stanley, R.,

Flag

f

-vectors and the

cd

-index

, Math. Z. 216 (1994), 483-499.

Title

𝐚𝐛

-index of graded posets

Canonical name

mathbfabindexOfGradedPosets

Date of creation

2013-03-22 15:46:47

Last modified on

2013-03-22 15:46:47

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Topic

Classification

msc 06A07

Synonym

ab-index

Synonym

cd-index

Synonym

𝐚𝐛

-index

Synonym

𝐜𝐝

-index

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