February

The

homology

of

partitions

with

an

even

number

of

blocks

Sheila

Department

of

Mathematics

and

Computer

Science

University

of

Miami

Coral

Gables

FL

Abstract

Let

denote

the

subposet

obtained

by

selecting

even

ranks

in

the

partition

lattice

We

show

that

the

homology

of

has

dimension

-LRB-

-RRB-

where

is

the

tangent

number

It

is

thus

an

integral

multiple

of

both

the

Genocchi

number

and

an

Andr?e

or

simsun

number

Using

the

general

theory

of

rank-selected

homology

representations

developed

in

-LSB-

Su

-RSB-

we

show

that

for

the

special

case

of

the

character

of

the

symmetric

group

on

the

homology

is

supported

on

the

set

of

involutions

Our

proof

techniques

lead

to

the

discovery

of

a

family

of

integers

bi

-LRB-

n

-RRB-

i

n

defined

recursively

We

conjecture

that

for

the

full

automorphism

group

the

homology

is

a

sum

of

permutation

modules

induced

from

Young

subgroups

of

the

form

with

nonnegative

integer

multiplicity

bi

-LRB-

n

-RRB-

The

nonnegativity

of

the

integers

bi

-LRB-

n

-RRB-

would

imply

the

existence

of

new

refinements

into

sums

of

powers

of

of

the

tangent

number

and

the

Andr?e

or

simsun

number

an

-LRB-

-RRB-

Similarly

the

restriction

of

this

homology

module

to

yields

a

family

of

integers

di

-LRB-

n

-RRB-

i

n

such

that

the

numbers

idi

-LRB-

n

-RRB-

refine

the

Genocchi

number

We

conjecture

that

idi

-LRB-

n

-RRB-

is

a

positive

integer

for

all

i

Finally

we

present

a

recursive

algorithm

to

generate

a

family

of

polynomials

which

encode

the

homology

representations

of

the

subposets

obtained

by

selecting

the

top

k

ranks

of

k

n

We

conjecture

that

these

are

all

permutation

modules

for

This

research

was

begun

while

the

author

was

on

leave

during

the

academic

year

The

author

gratefully

acknowledges

support

from

LACIM

Universit?e

du

Qu?ebec

a

Montr?eal

Montr?eal

Qu?ebec

Canada

for

the

Fall

of

and

from

the

Institut

Mittag-Leffler

Aurav?agen

Djursholm

Sweden

for

the

Spring

of

Introduction

Let

en

denote

the

subposet

of

the

partition

lattice

n

consisting

of

those

set

partitions

of

ng

which

have

an

even

number

of

blocks

This

subposet

may

also

be

obtained

by

selecting

alternate

ranks

in

the

partition

lattice

It

is

well

known

that

rank-selected

subposets

of

a

Cohen-Macaulay

poset

are

also

CohenMacaulay

-LRB-

see

-LSB-

Theorem

-RSB-

-LSB-

Mu

Corollary

-RSB-

and

-LSB-

Theorem

-RSB-

-RRB-

Since

n

is

Cohen-Macaulay

the

subposet

en

is

also

In

general

en

is

not

a

lattice

In

this

paper

we

study

the

representation

of

the

symmetric

group

Sn

on

the

homology

of

the

subposet

en

A

systematic

study

of

the

homology

representations

of

rank-selected

and

other

Cohen-Macaulay

subposets

of

n

was

initiated

in

-LSB-

Su

-RSB-

where

a

number

of

tools

were

introduced

We

shall

use

these

techniques

to

show

that

when

n

is

even

the

homology

representation

of

en

has

remarkable

properties

The

formulas

in

-LSB-

Su

Theorem

-RSB-

can

be

programmed

using

Stembridge

's

symmetric

functions

package

for

Maple

and

first

allowed

us

to

compute

the

homology

representations

This

data

overwhelmingly

supports

the

surprising

conjecture

that

the

-LRB-

reduced

-RRB-

top

homology

is

a

permutation

module

for

the

full

automorphism

group

Sn

when

n

is

even

Our

approach

to

this

problem

begins

with

a

recurrence

derived

in

-LSB-

Su

-RSB-

for

the

Frobenius

characteristics

of

these

representations

The

exposition

of

the

representation-theoretic

results

in

this

paper

is

cast

entirely

in

the

language

of

symmetric

functions

as

in

-LSB-

M

-RSB-

Consequently

no

attempt

is

made

to

define

homological

concepts

For

readers

who

have

some

knowledge

of

symmetric

functions

the

contents

of

this

paper

may

be

summarised

briefly

as

follows

We

begin

with

a

family

of

symmetric

functions

each

homogeneous

of

even

degree

defined

only

by

means

of

a

plethystic

recurrence

-LRB-

Theorem

-RRB-

By

means

of

a

purely

computational

and

rather

unilluminating

argument

we

are

able

to

show

-LRB-

Theorem

-RRB-

that

the

symmetric

function

is

in

fact

a

polynomial

with

integer

coefficients

in

the

homogeneous

symmetric

functions

and

Extensive

data

indicates

that

these

integers

are

nonnegative

and

representation-theoretic

considerations

suggest

that

these

integers

somehow

encode

mysterious

refinements

into

powers

of

of

several

well

known

numbers

such

as

the

tangent

number

The

main

result

of

Section

is

the

computation

of

the

M?obius

number

of

We

also

discuss

some

interesting

enumerative

aspects

of

this

number

In

Section

of

the

paper

we

examine

the

plethystic

recurrence

for

the

Frobenius

characteristic

of

the

homology

representation

of

more

closely

The

plethystic

generating

function

yields

valuable

information

on

the

representation

we

prove

that

the

character

values

vanish

outside

the

set

of

involutions

We

show

that

the

representation

is

completely

determined

by

a

two-variable

polynomial

with

integer

coefficients

by

the

simple

condition

that

all

terms

of

even

degree

within

a

given

range

vanish

identically

This

eliminates

the

need

for

plethystic

calculations

and

allows

us

to

compute

explicitly

the

data

supporting

our

conjecture

on

the

nature

of

the

homology

module

In

fact

we

present

a

recursive

algorithm

to

compute

a

more

general

polynomial

-LRB-

x

y

-RRB-

with

integer

coefficients

which

determines

for

m

such

that

m

-LRB-

the

symmetric

functions

encoding

-RRB-

the

homology

representation

of

em

as

well

as

that

of

the

subposet

of

em

consisting

of

the

top

k

ranks

The

results

of

this

computation

support

the

stronger

conjecture

that

the

homology

of

these

subposets

of

is

also

a

permutation

module

for

for

all

k

n

The

algorithm

has

allowed

us

to

verify

this

for

An

equivalent

formulation

of

our

conjecture

is

that

the

polynomial

-LRB-

x

y

-RRB-

has

nonnegative

coefficients

The

truth

of

our

conjecture

also

implies

that

the

homology

modules

of

these

posets

are

permutation

modules

for

the

subgroup

This

is

a

less

surprising

phenomenon

in

the

partition

lattice

see

-LSB-

Corollary

-RSB-

and

more

generally

-LSB-

Su

Theorem

-RSB-

Finally

in

Section

we

outline

the

enumerative

implications

of

our

representation-theoretic

conjecture

The

techniques

we

use

to

analyse

the

homology

representation

lead

to

the

discovery

of

a

family

of

integers

bi

-LRB-

n

-RRB-

i

n

for

which

we

give

an

explicit

but

unenlightening

recurrence

The

integer

bi

-LRB-

n

-RRB-

may

also

be

defined

as

the

-LRB-

virtual

-RRB-

multiplicity

of

the

permutation

module

induced

from

the

Young

subgroup

in

the

homology

module

H

-LRB-

-RRB-

Our

representation-theoretic

conjecture

is

equivalent

to

the

nonnegativity

of

the

integers

bi

-LRB-

n

-RRB-

If

true

this

conjecture

would

yield

apparently

new

refinements

of

the

tangent

number

the

Genocchi

number

and

certain

Andr?e

or

simsun

permutations

A

formula

for

computing

the

homology

representations

of

rank-selected

subposets

of

n

was

given

in

-LSB-

Su

Theorem

-RSB-

It

is

curious

that

of

all

the

proper

rank-selected

subposets

of

m

the

class

studied

in

this

paper

seems

to

be

the

only

one

whose

homology

is

interesting

as

an

Sm-module

In

particular

there

seems

to

be

a

special

significance

to

the

selection

of

even

ranks

A

similar

significance

seems

to

be

attached

to

the

selection

of

alternate

ranks

in

an

Eulerian

poset

P

with

nonnegative

cd-index

see

-LSB-

Corollary

-RSB-

One

might

ask

if

there

are

other

approaches

to

showing

that

a

homology

module

is

a

permutation

module

In

the

case

of

posets

which

are

homotopy

equivalent

to

a

wedge

of

spheres

one

can

argue

that

if

the

homology

spheres

themselves

are

permuted

by

the

group

action

then

there

must

be

an

element

of

the

poset

which

is

fixed

by

the

group

action

namely

an

element

belonging

to

the

facet

which

-LRB-

eventually

by

contraction

-RRB-

serves

as

a

common

wedge

point

for

the

bouquet

of

spheres

in

homology

Clearly

no

partition

in

is

fixed

by

and

hence

one

can

not

hope

to

find

a

basis

of

fundamental

cycles

in

homology

-LRB-

or

dually

a

basis

of

cochains

in

cohomology

-RRB-

which

is

permuted

by

Two

other

instances

when

a

-LRB-

co

-RRB-

homology

representation

turns

out

to

be

a

permutation

module

for

the

full

automorphism

group

have

recently

appeared

in

the

algebraic

combinatorics

literature

One

is

the

classical

case

of

the

cohomology

of

the

flag

variety

-LRB-

see

-LSB-

M

p.

Example

-RSB-

-RRB-

more

recently

studied

from

a

combinatorial

viewpoint

by

Garsia

and

Procesi

The

other

example

arises

most

recently

in

the

work

of

Stembridge

-LSB-

Ste

-RSB-

who

shows

that

the

cohomology

of

the

toric

variety

associated

to

the

Coxeter

complex

of

a

crystallographic

root

system

is

always

a

permutation

module

for

the

corresponding

Weyl

group

In

both

cases

the

cohomology

is

a

graded

version

of

a

permutation

representation

the

graded

components

themselves

are

not

always

permutation

modules

The

proofs

in

both

the

above

cases

use

formal

rather

than

constructive

methods

indeed

in

neither

instance

does

it

seem

to

be

known

how

to

exhibit

a

basis

that

is

actually

permuted

by

the

group

The

isotropy

groups

of

the

induced

modules

appearing

in

this

paper

will

always

be

Young

subgroups

Hence

notation

such

as

Sa

Sb

or

Smn

Sn

Sn

-LCB-

z

-RCB-

m

copies

will

always

refer

to

the

appropriate

Young

subgroups

in

Sa

b

and

Smn

respectively

Betti

numbers

Denote

by

n

the

absolute

value

of

the

M?obius

number

of

the

poset

en

By

the

Cohen-Macaulay

property

of

the

poset

this

is

also

the

rank

of

the

unique

nonzero

homology

group

over

the

integers

that

is

the

Betti

number

of

the

simplicial

complex

of

chains

of

the

poset

Thus

-LRB-

-RRB-

n

-LRB-

-RRB-

We

shall

use

the

classical

compositional

formula

as

stated

in

-LSB-

-RSB-

to

compute

Theorem

-LRB-

-LSB-

Theorem

-RSB-

-RRB-

Let

K

be

a

field

of

characteristic

zero

and

let

f

and

g

be

K

valued

functions

defined

on

the

set

of

nonnegative

integers

with

f

-LRB-

-RRB-

and

with

respective

exponential

generating

functions

Ef

Eg

Define

a

K-valued

function

h

on

the

nonnegative

integers

by

h

-LRB-

n

-RRB-

ae

P

ss

Bkg

f

-LRB-

-RRB-

f

-LRB-

-RRB-

f

-LRB-

jBkj

-RRB-

g

-LRB-

k

-RRB-

if

n

if

n

where

the

sum

ranges

over

all

set

partitions

ss

Bkg

of

a

set

of

size

n

into

nonempty

blocks

Then

the

exponential

generating

function

for

h

is

given

by

Eh

-LRB-

x

-RRB-

Eg

-LRB-

Ef

-LRB-

x

-RRB-

-RRB-

Proposition

For

k

define

g

-LRB-

k

-RRB-

to

be

if

k

is

odd

and

-LRB-

-RRB-

n

if

k

is

even

in

particular

g

-LRB-

-RRB-

Set

B

-LRB-

x

-RRB-

Eg

-LRB-

x

-RRB-

P

n

-LRB-

-RRB-

n

-LRB-

-RRB-

Then

B

-LRB-

x

-RRB-

satisfies

the

functional

equation

B

-LRB-

x

-RRB-

B

-LRB-

x

x

-RRB-

x

Proof

We

apply

the

compositional

formula

to

the

functions

g

and

f

such

that

f

-LRB-

n

-RRB-

for

all

n

f

-LRB-

-RRB-

Note

that

Ef

-LRB-

x

-RRB-

ex

Letting

S

-LRB-

m

k

-RRB-

denote

as

usual

the

Stirling

number

of

the

second

kind

-LRB-

which

counts

the

number

of

set

partitions

in

m

with

k

blocks

-RRB-

we

find

that

h

-LRB-

-RRB-

Pnk

S

-LRB-

-RRB-

-LRB-

-RRB-

k

since

the

sum

in

Theorem

must

now

range

over

all

partitions

with

an

even

number

of

blocks

Observe

that

the

top

element

denoted

by

has

to

be

artificially

adjoined

to

Thus

h

-LRB-

-RRB-

X

-LRB-

x

-RRB-

which

in

turn

is

-LRB-

-RRB-

by

definition

of

the

M?obius

function

Similarly

h

-LRB-

-RRB-

n

X

k

S

-LRB-

-RRB-

-LRB-

-RRB-

k

X

-LRB-

x

-RRB-

-LRB-

-RRB-

since

now

the

top

and

the

bottom

element

-LRB-

-RRB-

have

to

be

artificially

adjoined

to

the

poset

Now

from

Theorem

using

the

expression

for

Eg

-LRB-

x

-RRB-

in

the

statement

of

the

proposition

we

obtain

-LRB-

ex

-RRB-

X

n

-LRB-

-RRB-

n

-LRB-

-RRB-

X

n

-LRB-

-RRB-

n

-LRB-

ex

-RRB-

-LRB-

-RRB-

-LRB-

A

-RRB-

Eliminating

the

odd

powers

of

x

in

this

identity

and

substituting

log

-LRB-

x

-RRB-

for

x

we

obtain

the

required

functional

equation

We

can

now

deduce

explicitly

the

value

of

Let

be

the

nth

tangent

number

n

-LRB-

that

is

tan

-LRB-

x

-RRB-

P

n

-LRB-

-RRB-

-RRB-

The

author

is

grateful

to

Richard

Stanley

and

Ira

Gessel

for

showing

her

the

following

elegant

technique

which

eliminated

the

cumbersome

calculations

of

her

original

proof

Define

a

map

on

the

ring

of

formal

power

series

with

coefficients

in

a

field

K

of

characteristic

zero

as

follows

-LRB-

X

n

f

-LRB-

n

-RRB-

xn

-RRB-

X

n

f

-LRB-

n

-RRB-

xn

n

Lemma

Let

F

-LRB-

x

-RRB-

P

n

f

-LRB-

n

-RRB-

xn

Under

the

map

the

image

of

the

power

series

ffxF

-LRB-

x

ffx

-RRB-

is

e?ffx

-LRB-

F

-LRB-

x

-RRB-

-RRB-

For

a

proof

see

-LSB-

Exercises

-LRB-

a

-RRB-

and

-LRB-

a

-RRB-

-RSB-

Theorem

The

top

homology

of

the

poset

is

a

space

of

dimension

-LRB-

-RRB-

Equivalently

this

is

the

value

of

-LRB-

-RRB-

n

times

the

M?obius

number

Proof

Write

-LRB-

-RRB-

n

-LRB-

-RRB-

Note

that

the

exponential

generating

function

for

the

numbers

-LRB-

-RRB-

is

i

tan

-LRB-

ix

-RRB-

where

Hence

it

suffices

to

show

that

the

generating

function

P

n

-LRB-

-RRB-

for

the

numbers

is

i

tan

-LRB-

-RRB-

tanh

-LRB-

-RRB-

Apply

Lemma

to

the

functional

equation

of

Propostion

taking

F

-LRB-

x

-RRB-

B

-LRB-

x

-RRB-

The

result

now

follows

by

applying

the

map

noting

that

-LRB-

x

-RRB-

e?x

Remark

Here

we

collect

some

identities

involving

the

Betti

numbers

-LRB-

-RRB-

n

-LRB-

-RRB-

-LRB-

i

-RRB-

If

we

use

the

defining

equation

for

the

M?obius

function

on

the

poset

-LSB-

we

obtain

-LRB-

with

-RRB-

the

identity

Pnk

-LRB-

-RRB-

kS

-LRB-

-RRB-

or

equivalently

from

the

preceding

result

n

X

k

-LRB-

-RRB-

k

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

ii

-RRB-

Recall

from

-LSB-

Example

-RSB-

that

an

atom-ordering

of

a

geometric

lattice

gives

an

R-labelling

-LRB-

in

fact

an

admissible

labelling

-RRB-

of

the

lattice

Theorem

of

-LSB-

-RSB-

then

says

that

the

Betti

number

of

a

rank-selected

subposet

of

the

lattice

is

given

by

the

number

of

chains

with

descents

in

the

positions

corresponding

to

the

ranks

selected

-LRB-

It

is

shown

in

-LSB-

-RSB-

that

an

admissible

labelling

of

a

lattice

is

in

fact

an

EL-labelling

and

consequently

defines

a

shelling

order

on

the

maximal

chains

of

the

lattice

-RRB-

The

atom-ordering

corresponding

to

Gessel

's

EL-labelling

for

the

partition

lattice

as

described

in

-LSB-

-RSB-

is

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

i

i

-RRB-

-LRB-

i

-RRB-

-LRB-

n

n

-RRB-

The

chains

in

with

descents

in

the

positions

corresponding

to

even

ranks

may

be

termed

up-down

chains

counting

up-down

chains

with

respect

to

this

labelling

yields

the

formula

-LRB-

-LSB-

ES

Theorem

-RSB-

-RRB-

X

oe

is

an

up?down

permutation

Y

k

-LRB-

jfi

i

k

oe

-LRB-

i

-RRB-

oe

-LRB-

k

-RRB-

gj

-RRB-

and

hence

by

Theorem

-LRB-

-RRB-

X

oe

is

an

up?down

permutation

Y

k

-LRB-

jfi

i

k

oe

-LRB-

i

-RRB-

oe

-LRB-

k

-RRB-

gj

-RRB-

-LRB-

iii

-RRB-

The

well

known

recurrence

n

X

i

for

the

tangent

number

translates

into

the

following

recurrence

for

the

Betti

number

n

X

i

It

is

unclear

to

us

how

this

latter

formula

can

be

explained

by

means

of

the

combinatorial

interpretation

of

up-down

chains

in

described

in

-LRB-

ii

-RRB-

-LRB-

iv

-RRB-

It

is

worth

noting

that

by

a

result

of

Stanley

-LSB-

-RSB-

the

tangent

number

is

itself

the

Betti

number

of

a

subposet

of

the

partition

lattice

namely

the

join

sublattice

of

generated

by

the

partitions

with

n

blocks

all

of

size

The

homology

representation

of

for

this

lattice

was

determined

in

-LSB-

CHR

-RSB-

-LRB-

v

-RRB-

Finally

the

Euler

number

Em

is

also

the

Betti

number

of

the

subposet

of

the

Boolean

lattice

Bm

consisting

of

alternate

ranks

The

homology

representation

of

Sm

in

this

case

was

computed

by

Solomon

-LRB-

-LSB-

So

-RSB-

-RRB-

to

be

one

of

the

Foulkes

representations

it

is

the

representation

of

Sm

indexed

by

a

skew-hook

or

border

strip

-LRB-

see

-LSB-

M

-RSB-

for

definitions

-RRB-

formed

by

horizontal

strips

all

of

size

with

the

exception

of

the

top-most

strip

which

may

be

of

size

Remark

From

the

proof

of

Proposition

the

Betti

numbers

of

the

poset

are

determined

by

equation

-LRB-

A

-RRB-

in

the

proof

We

are

unable

to

obtain

an

explicit

closed

formula

for

The

first

few

values

for

n

and

are

and

Next

we

present

some

numerical

evidence

which

suggests

the

following

The

numbers

are

related

to

the

number

of

maximal

chains

in

the

full

partition

lattice

which

are

orbit

representatives

of

chains

fixed

by

the

action

of

or

equivalently

representatives

of

orbits

of

size

-LRB-

-RRB-

This

in

turn

implies

a

connection

with

certain

types

of

Andr?e

permutations

-LSB-

FSch

-RSB-

In

-LSB-

Su

-RSB-

the

action

of

Sn

on

the

maximal

chains

of

n

was

studied

and

the

following

theorem

was

proved

Define

positive

integers

ai

-LRB-

n

-RRB-

by

means

of

the

recurrence

ai

-LRB-

n

-RRB-

iai

-LRB-

n

-RRB-

-LRB-

n

-RRB-

ai

-LRB-

n

-RRB-

with

initial

conditions

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

n

-RRB-

n

and

ai

-LRB-

n

-RRB-

if

n

We

have

Theorem

-LRB-

-LSB-

Su

Theorem

-RSB-

-RRB-

The

action

of

Sn

on

the

chains

of

n

decomposes

into

orbits

as

follows

b

c

X

i

ai

-LRB-

n

-RRB-

Wi

where

Wi

denotes

the

transitive

permutation

representation

of

Sn

acting

on

the

cosets

of

Sn

It

follows

from

-LSB-

Theorem

-RSB-

that

the

positive

integers

ai

-LRB-

n

-RRB-

refine

the

Euler

number

En

R.

Simionand

this

author

observed

from

the

above

recurrence

that

the

numbers

ai

-LRB-

n

-RRB-

admit

the

following

combinatorial

interpretation

ai

-LRB-

n

-RRB-

is

the

number

of

permutations

in

Sn

with

exactly

i

descents

consecutive

and

with

the

hereditary

property

that

when

the

letters

n

n

are

erased

in

succession

the

property

of

not

having

any

consecutive

descents

is

preserved

after

each

erasure

We

call

these

simsun

permutations

The

equinumerous

set

of

Andr?e

permutations

appears

in

work

of

Foata

and

Sch?utzenberger

-LRB-

see

-LSB-

V

-RSB-

and

also

the

references

in

-LSB-

Su

-RSB-

-RRB-

Both

sets

reappear

in

recent

work

of

Purtill

and

Stanley

on

the

cd-index

-LRB-

see

-LSB-

-RSB-

-RRB-

With

the

Foata-Sch

utzenberger

interpretation

ai

-LRB-

n

-RRB-

is

the

number

of

Andr?e

permutations

in

Sn

with

exactly

i

peaks

where

j

is

a

peak

if

either

j

and

oe

-LRB-

-RRB-

oe

-LRB-

-RRB-

or

if

j

n

and

oe

-LRB-

j

-RRB-

max

-LRB-

oe

-LRB-

j

-RRB-

oe

-LRB-

j

-RRB-

-RRB-

Note

in

particular

that

an

-LRB-

-RRB-

is

the

number

of

alternating

-LRB-

necessarily

down-up

-RRB-

permutations

in

with

the

hereditary

property

of

no

consecutive

descents

described

above

It

now

transpires

that

the

Betti

number

is

an

exact

multiple

of

the

number

an

-LRB-

-RRB-

Proposition

We

have

-LRB-

-RRB-

and

-LRB-

-RRB-

an

-LRB-

-RRB-

Proof

From

Theorem

we

see

that

an

-LRB-

-RRB-

is

the

number

of

orbits

of

maximal

saturated

chains

in

which

are

fixed

point-wise

under

the

action

of

a

subgroup

conjugate

to

In

order

that

a

partition

be

fixed

by

all

involutions

generated

by

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

it

must

be

the

case

that

either

and

are

in

the

same

block

or

they

are

both

singletons

Hence

an

-LRB-

-RRB-

is

the

number

of

orbits

of

chains

in

the

subposet

of

partitions

with

the

property

that

the

only

blocks

of

odd

size

are

the

singletons

and

the

latter

occur

in

pairs

We

use

this

fact

and

a

counting

argument

completely

analogous

to

-LSB-

Theorem

-RSB-

to

obtain

the

recurrence

an

-LRB-

-RRB-

n

X

i

ai

-LRB-

-RRB-

an?i

-LRB-

-RRB-

To

see

this

first

choose

the

element

of

maximal

rank

in

the

representative

chain

such

an

element

is

of

the

form

ss

where

and

are

blocks

of

sizes

and

respectively

i

n

Since

we

are

counting

orbit

representatives

it

is

enough

to

choose

one

pair

of

blocks

for

each

pair

of

sizes

Now

choose

representative

chains

from

in

ai

-LRB-

-RRB-

ways

and

from

in

an?i

-LRB-

-RRB-

ways

these

can

be

intertwined

in

ways

to

produce

a

maximal

chain

in

Now

from

-LRB-

iii

-RRB-

of

Remark

it

is

clear

that

this

coincides

with

the

recurrence

satisfied

by

the

numbers

Remark

Let

cn

denote

the

number

of

maximal

saturated

chains

in

n

Counting

from

the

bottom

we

have

cn

n

-LRB-

n

-RRB-

We

then

have

-LRB-

see

-LSB-

Su

Remark

-RSB-

-RRB-

-LRB-

n

-RRB-

c

i

ai

-LRB-

n

-RRB-

i

this

decomposition

is

explained

combinatorially

in

-LSB-

-RSB-

and

-LSB-

-RSB-

by

means

of

an

action

of

the

subgroup

Sb

n

c

on

the

set

of

-LRB-

n

-RRB-

permutations

See

also

-LSB-

V

-RSB-

It

would

be

interesting

to

investigate

combinatorially

the

various

enumerative

formulas

of

this

section

The

homology

representation

It

is

instructive

to

look

at

the

first

nontrivial

example

Here

the

elements

are

all

at

rank

there

are

four

partitions

in

the

orbit

of

and

three

in

the

orbit

of

The

reduced

homology

is

concentrated

in

dimension

zero

and

as

an

it

is

a

sum

of

two

transitive

permutation

representations

with

one

copy

of

the

trivial

representation

removed

A

little

thought

reveals

the

following

surprising

fact

the

homology

module

is

simply

the

-LRB-

single

-RRB-

transitive

permutation

module

induced

from

the

Young

subgroup

-LRB-

although

as

observed

in

the

introduction

this

is

not

obvious

from

any

basis

of

fundamental

cycles

-RRB-

This

is

our

first

hint

that

the

homology

modules

of

have

interesting

properties

We

refer

to

-LSB-

M

-RSB-

for

all

background

on

symmetric

functions

As

in

-LSB-

M

-RSB-

we

denote

the

plethysm

of

symmetric

functions

f

g

by

f

-LSB-

g

-RSB-

If

f

and

g

are

respectively

Frobenius

characteristics

of

the

Sm-module

V

and

the

Sn-module

W

then

the

plethysm

f

-LSB-

g

-RSB-

is

the

characteristic

of

the

module

V

-LSB-

W

-RSB-

induced

up

to

Smn

from

the

wreath

product

subgroup

Sm

-LSB-

Sn

-RSB-

where

Sm

acts

on

m

copies

of

Sn

-LRB-

Note

that

Sm

-LSB-

Sn

-RSB-

is

the

normaliser

in

Smn

of

the

Young

subgroup

Sn

Sn

-LCB-

z

-RCB-

m

copies

:-RRB-

The

remainder

of

this

section

is

devoted

to

studying

the

symmetric

functions

associated

to

the

homology

representations

via

the

Frobenius

characteristic

map

We

begin

with

a

family

of

symmetric

functions

Rn

For

the

purposes

of

this

paper

the

plethystic

recurrence

of

Theorem

below

may

be

taken

to

be

the

defining

equation

for

the

functions

Rn

-LRB-

We

note

in

passing

that

because

Rn

is

the

Frobenius

characteristic

of

a

representation

we

know

in

addition

that

it

is

a

nonnegative

integer

combination

of

Schur

functions

-RRB-

Theorem

-LRB-

See

-LSB-

Su

Theorem

-RSB-

-RRB-

Let

Rn

denote

the

Frobenius

characteristic

of

the

representation

of

Sn

on

the

top

homology

of

en

Then

-LRB-

i

-RRB-

X

n

-LRB-

-RRB-

-LRB-

:-RRB-

-LSB-

Equivalently

separating

terms

of

even

and

odd

degree

we

find

that

-LRB-

respectively

-LRB-

-RRB-

n

-RRB-

is

the

degree

-LRB-

-RRB-

-LRB-

respectively

degree

-LRB-

-RRB-

-RRB-

term

in

the

alternating

sum

of

plethysms

-LRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

n

-LRB-

-RRB-

n

-RRB-

j

hj

while

terms

in

this

sum

of

even

degree

smaller

than

-LRB-

-RRB-

vanish

identically

-LRB-

ii

-RRB-

Let

em

-LRB-

k

-RRB-

denote

the

rank-selected

subposet

of

em

consisting

of

the

top

k

nontrivial

ranks

fn

n

kg

k

n

for

m

-LRB-

In

particular

if

k

n

this

is

the

whole

poset

em

-RRB-

Then

the

Frobenius

characteristic

of

the

top

homology

of

this

subposet

as

an

Sm-module

is

given

by

the

degree

m

term

in

the

alternating

sum

-LRB-

-LRB-

k

-RRB-

-LRB-

-RRB-

k

-LRB-

-RRB-

-RRB-

j

hj

In

this

section

we

shall

use

the

defining

equation

above

to

investigate

the

representations

The

following

well

known

expansion

of

the

plethysm

-LSB-

hn

-RSB-

into

Schur

functions

is

easily

established

by

routine

computation

It

also

provides

an

explanation

for

the

phenomenon

noticed

at

the

beginning

of

this

section

for

the

representation

Proposition

-LSB-

hn

-RSB-

c

k

s

-LRB-

-RRB-

Now

write

He

for

the

sum

P

n

Ho

P

n

and

H

He

Ho

The

following

identity

is

the

essential

trick

to

simplifying

the

plethystic

recurrence

and

is

the

crucial

ingredient

in

the

proof

of

Theorem

below

Proposition

-LSB-

H

-RSB-

He

-LRB-

H

-RRB-

He

-LRB-

He

-RRB-

HoHe

Proof

From

the

rules

for

computing

plethysm

of

symmetric

functions

we

have

-LSB-

H

-RSB-

P

n

-LSB-

hn

-RSB-

P

n

P

i

i

n?i

hihn?i

Using

Proposition

the

result

follows

The

next

observation

will

be

used

frequently

Lemma

The

lowest

degree

term

in

o

is

of

degree

and

equals

We

are

now

ready

to

establish

the

following

surprising

result

Theorem

The

Frobenius

characteristic

is

a

polynomial

in

the

homogeneous

symmetric

functions

and

Equivalently

the

character

values

of

acting

on

the

top

homology

of

are

supported

on

involutions

Proof

We

shall

proceed

by

induction

on

n

Note

first

that

using

the

defining

equation

we

have

since

-LSB-

H

-RSB-

He

Ho

Now

is

the

degree

term

in

-LRB-

-RRB-

-LSB-

H

-RSB-

that

is

-LRB-

using

Propostion

-RRB-

the

degree

term

in

Ho

-LRB-

He

-RRB-

Hence

Our

induction

hypothesis

will

be

that

the

plethysm

-LRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

n

-LRB-

-RRB-

-RRB-

-LSB-

H

-RSB-

is

a

polynomial

in

He

and

Ho

with

integer

coefficients

More

specifically

isolating

the

terms

of

even

and

odd

degree

with

respect

to

the

grading

deg

-LRB-

He

-RRB-

deg

-LRB-

Ho

-RRB-

we

assume

that

there

exist

polynomials

-LRB-

n

-RRB-

-LRB-

x

y

-RRB-

and

-LRB-

n

-RRB-

-LRB-

x

y

-RRB-

in

the

variables

x

and

y

with

integer

coefficients

such

that

-LRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

n

-LRB-

-RRB-

-RRB-

-LSB-

H

-RSB-

-LRB-

n

-RRB-

-LRB-

He

-RRB-

-LRB-

n

-RRB-

-LRB-

He

-RRB-

-LRB-

A

-RRB-

we

further

assume

that

-LRB-

n

-RRB-

-LRB-

x

y

-RRB-

is

divisible

by

for

n

-LRB-

Note

in

particular

that

with

respect

to

the

above

grading

-LRB-

n

-RRB-

-LRB-

He

-RRB-

and

-LRB-

n

-RRB-

-LRB-

He

-RRB-

are

both

of

even

degree

-RRB-

From

the

first

paragraph

of

the

proof

we

have

-LRB-

He

-RRB-

He

-LRB-

He

-RRB-

and

-LRB-

He

-RRB-

-LRB-

He

-RRB-

He

hence

these

hypotheses

are

satisfied

for

n

Note

that

by

-LRB-

A

-RRB-

only

the

terms

in

-LRB-

n

-RRB-

-LRB-

He

-RRB-

contribute

to

The

defining

equation

of

Theorem

-LRB-

i

-RRB-

implies

that

is

the

degree

term

in

-LRB-

n

-RRB-

-LRB-

He

-RRB-

by

Lemma

since

all

terms

of

smaller

even

degree

in

the

left-hand

side

of

-LRB-

A

-RRB-

must

vanish

identically

this

is

the

lowest

degree

term

in

-LRB-

n

-RRB-

-LRB-

He

-RRB-

as

a

polynomial

in

He

and

Now

let

n

By

induction

hypothesis

the

lowest

degree

term

in

-LRB-

He

-RRB-

is

of

the

form

n

X

i

bi

-LRB-

n

-RRB-

Hie

-LRB-

-RRB-

n?i

for

some

integers

bi

-LRB-

n

-RRB-

i

n

By

Lemma

equals

Pni

bi

-LRB-

n

-RRB-

and

hence

using

Proposition

-LSB-

H

-RSB-

equals

n

X

i

bi

-LRB-

n

-RRB-

-LRB-

-LSB-

H

-RSB-

-RRB-

n

X

i

bi

-LRB-

n

-RRB-

Hie

-LRB-

He

Ho

-RRB-

i

-LRB-

He

Ho

-RRB-

-LRB-

B

-RRB-

In

particular

-LSB-

H

-RSB-

is

divisible

by

It

follows

that

there

are

polynomials

-LRB-

x

y

-RRB-

and

-LRB-

x

y

-RRB-

with

integer

coefficients

such

that

-LRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

n

-LRB-

-RRB-

n

-RRB-

-LSB-

H

-RSB-

-LRB-

He

-RRB-

-LRB-

He

-RRB-

-LRB-

C

-RRB-

where

by

inspecting

the

right-hand

side

of

-LRB-

B

-RRB-

since

-LRB-

He

-RRB-

even

degree

terms

in

-LRB-

He

-RRB-

-LRB-

He

-RRB-

it

is

clear

that

-LRB-

He

-RRB-

is

divisible

by

Note

also

that

with

respect

to

the

grading

deg

-LRB-

He

-RRB-

deg

-LRB-

Ho

-RRB-

the

lowest

degree

term

in

-LRB-

B

-RRB-

coincides

with

the

lowest

degree

term

in

-LRB-

He

-RRB-

and

has

degree

Hence

the

lowest

degree

term

in

-LRB-

He

-RRB-

has

degree

-LRB-

-RRB-

and

is

of

the

form

n

X

i

bi

-LRB-

n

-RRB-

Hie

-LRB-

-RRB-

n

i

-LRB-

D

-RRB-

for

some

integers

bi

-LRB-

n

-RRB-

i

n

It

now

follows

by

another

application

of

Lemma

that

-LRB-

n

-RRB-

which

by

the

defining

equation

is

the

term

of

degree

-LRB-

n

-RRB-

in

the

symmetric

function

-LRB-

He

-RRB-

is

of

the

form

n

X

i

bi

-LRB-

n

-RRB-

-LRB-

n

-RRB-

where

the

bi

-LRB-

n

-RRB-

are

integers

uniquely

determined

from

-LRB-

A

-RRB-

-LRB-

B

-RRB-

and

-LRB-

C

-RRB-

The

second

assertion

in

the

statement

of

the

theorem

follows

from

the

fact

that

the

character

of

the

permutation

representation

on

cosets

of

the

Young

subgroup

is

supported

on

the

set

of

involutions

Now

let

n

and

let

bi

-LRB-

n

-RRB-

be

the

unique

integers

such

that

Pni

bi

-LRB-

n

-RRB-

Remark

Converting

the

homogeneous

symmetric

functions

to

power-sums

we

see

that

the

character

of

the

homology

representation

on

involutions

of

type

-LRB-

-RRB-

is

given

by

n

X

j

bj

-LRB-

n

-RRB-

j

i

-LRB-

-RRB-

In

particular

the

character

value

on

fixed-point-free

involutions

is

bn

-LRB-

n

-RRB-

n

Let

-LRB-

x

y

-RRB-

be

the

polynomial

defined

recursively

by

x

and

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

-LRB-

-RRB-

x

-LRB-

x

y

-RRB-

n

X

k

-LRB-

-RRB-

-LRB-

x

x

-LRB-

x

y

-RRB-

y

x

y

-RRB-

-LRB-

x

y

-RRB-

x

-LRB-

x

y

-RRB-

n

X

k

-LRB-

-RRB-

k

k

X

i

bi

-LRB-

k

-RRB-

xi

-LRB-

x

y

-RRB-

i

-LRB-

x

y

-RRB-

We

have

Corollary

With

respect

to

the

grading

deg

x

deg

y

all

terms

of

total

-LRB-

even

-RRB-

degree

at

most

in

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

vanish

identically

and

the

resulting

equations

determine

the

bi

-LRB-

n

-RRB-

for

n

This

gives

the

recurrences

-LRB-

n

-RRB-

and

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

n

-RRB-

-LRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-RRB-

Proof

Define

-LRB-

-RRB-

Then

-LRB-

-RRB-

The

defining

equation

of

Theorem

implies

that

all

terms

of

even

degree

at

most

vanish

in

the

symmetric

functions

expansion

of

-LSB-

H

-RSB-

-LSB-

H

-RSB-

-LRB-

-RRB-

-LSB-

H

-RSB-

Using

Proposition

we

find

that

all

terms

of

even

degree

at

most

in

the

symmetric

function

expansion

of

-LRB-

He

Ho

-RRB-

-LRB-

-RRB-

He

-LRB-

He

Ho

-RRB-

n

X

k

-LRB-

-RRB-

k

k

X

i

bi

-LRB-

k

-RRB-

Hie

-LRB-

He

Ho

-RRB-

i

-LRB-

He

Ho

-RRB-

must

vanish

identically

Recall

that

He

-LRB-

respectively

Ho

-RRB-

contributes

only

even

-LRB-

respectively

odd

-RRB-

degree

terms

Write

-LRB-

x

y

-RRB-

for

the

polynomial

obtained

by

substituting

x

for

He

and

y

for

Ho

i.e.

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

x

-LRB-

x

y

-RRB-

n

X

k

-LRB-

-RRB-

k

k

X

i

bi

-LRB-

k

-RRB-

xi

-LRB-

x

y

-RRB-

i

-LRB-

x

y

-RRB-

The

defining

equation

for

the

bi

-LRB-

n

-RRB-

now

translates

into

the

statement

that

with

respect

to

the

grading

deg

x

deg

y

all

terms

of

total

even

degree

at

most

in

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

vanish

identically

In

particular

extracting

the

coefficient

of

in

-LRB-

x

y

-RRB-

we

find

that

-LRB-

k

-RRB-

-LRB-

k

-RRB-

for

all

k

Since

-LRB-

-RRB-

it

follows

that

-LRB-

k

-RRB-

for

all

k

Similarly

extracting

the

coefficient

of

we

obtain

-LRB-

k

-RRB-

-LRB-

k

-RRB-

-LRB-

-RRB-

-LRB-

k

-RRB-

The

recurrences

of

Corollary

imply

that

bi

-LRB-

n

-RRB-

for

i

and

To

see

that

-LRB-

n

-RRB-

is

positive

note

first

that

-LRB-

n

-RRB-

-LRB-

n

-RRB-

for

all

n

Assume

inductively

that

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

This

is

true

for

n

:-RRB-

Then

the

recurrence

shows

that

-LRB-

n

-RRB-

-LRB-

n

-RRB-

for

all

n

Remark

The

general

recurrence

for

the

bi

-LRB-

n

-RRB-

can

be

extracted

from

this

corollary

and

is

recorded

in

Conjecture

of

Section

The

proof

of

Theorem

yields

an

efficient

way

to

compute

the

characteristic

without

going

through

laborious

plethystic

calculations

It

is

enough

to

extract

the

coefficient

of

in

the

polynomial

-LRB-

-RRB-

We

obtain

up

to

n

These

computations

lead

us

to

make

the

following

Conjecture

The

homology

of

is

in

fact

a

permutation

module

for

it

is

a

sum

of

induced

modules

whose

isotropy

groups

are

Young

subgroups

of

the

form

with

nonnegative

integer

multiplicity

bi

-LRB-

n

-RRB-

i

n

The

integers

bi

-LRB-

n

-RRB-

are

defined

as

in

Corollary

It

is

also

clear

from

Theorem

that

may

be

viewed

as

a

sum

of

-LRB-

possibly

virtual

-RRB-

representations

induced

from

the

Young

subgroup

In

fact

we

have

a

second

way

of

writing

as

such

Corollary

For

i

n

let

Vi

be

the

irreducible

representation

of

which

acts

like

the

trivial

representation

on

i

copies

of

and

like

the

sign

on

the

remaining

-LRB-

n

i

-RRB-

copies

Define

integers

Ek

-LRB-

n

-RRB-

by

Ek

-LRB-

n

-RRB-

Pni

bi

-LRB-

n

-RRB-

n?i

k?i

k

n

Then

one

has

a

-LRB-

possibly

virtual

-RRB-

direct

sum

decomposition

H

-LRB-

-RRB-

n

X

i

Ei

-LRB-

n

-RRB-

Vi

In

particular

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

-RRB-

and

En

-LRB-

n

-RRB-

is

the

multiplicity

of

the

trivial

representation

in

the

top

homology

of

H

-LRB-

-RRB-

Proof

Make

the

substitution

in

the

characteristic

It

follows

that

Conjecture

if

true

would

also

imply

the

positivity

of

the

integers

Ei

-LRB-

n

-RRB-

We

already

know

that

this

is

the

case

for

En

-LRB-

n

-RRB-

since

it

is

the

multiplicity

of

the

trivial

representation

in

H

-LRB-

-RRB-

The

first

few

values

of

the

Ei

-LRB-

n

-RRB-

up

to

n

are

given

below

as

the

coefficient

of

the

characteristic

of

the

module

Vi

of

Corollary

Finally

Theorem

can

also

be

used

to

compute

the

Whitney

homology

of

the

dual

of

combined

with

Proposition

of

-LSB-

Su

-RSB-

one

can

then

compute

the

homology

of

the

rank

k

subposets

-LRB-

k

-RRB-

obtained

by

selecting

the

top

k

nontrivial

ranks

of

-LRB-

k

n

-RRB-

We

have

the

following

particular

cases

Proposition

-LRB-

i

-RRB-

The

characteristic

of

-LRB-

-LRB-

-RRB-

-RRB-

is

equal

to

X

k

n?k

even

where

ffin

even

equals

if

n

is

even

and

otherwise

-LRB-

ii

-RRB-

If

is

an

integer

partition

of

n

denote

by

-LRB-

-RRB-

the

number

of

parts

of

Also

write

mi

-LRB-

-RRB-

for

the

multiplicity

of

the

part

i

in

The

characteristic

of

-LRB-

-LRB-

-RRB-

-RRB-

is

equal

to

X

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

h

X

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

h

n

X

k

X

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

h?h

where

the

first

and

second

sums

run

over

partitions

with

all

parts

even

and

the

third

sum

runs

over

pairs

of

partitions

-LRB-

-RRB-

such

that

all

the

parts

of

are

even

and

all

the

parts

of

are

odd

Proof

-LRB-

i

-RRB-

From

Theorem

-LRB-

-RRB-

the

characteristic

is

the

degree

term

in

-LSB-

H

-RSB-

H

He

-LRB-

He

Ho

-RRB-

H

hence

equals

the

degree

term

in

-LRB-

ii

-RRB-

This

time

the

characteristic

is

the

degree

term

in

-LSB-

H

-RSB-

-LSB-

H

-RSB-

H

-LRB-

He

Ho

-RRB-

He

-LRB-

He

Ho

-RRB-

H

hence

equals

the

degree

term

in

-LRB-

-RRB-

The

results

follow

Notice

that

both

homology

modules

of

Proposition

are

permutation

modules

for

This

observation

is

reinforced

by

data

from

further

computations

We

can

now

strengthen

Conjecture

by

the

following

representation-theoretic

conjecture

Conjecture

The

homology

Hk

-LRB-

-LRB-

k

-RRB-

-RRB-

for

k

n

is

a

permutation

module

for

the

action

of

and

can

be

written

as

a

sum

of

induced

modules

whose

point

stabilisers

are

all

Young

subgroups

In

general

we

are

unable

to

discern

any

particular

distinguishing

qualities

for

the

isotropy

subgroups

beyond

the

fact

that

Young

subgroups

indexed

by

the

conjugacy

classes

of

involutions

appear

only

for

k

n

Remark

For

the

case

k

we

can

describe

a

basis

for

the

-LRB-

co

-RRB-

homology

which

is

permuted

by

the

group

action

For

each

i

such

that

i

n

let

oi

be

the

partition

whose

blocks

are

ig

and

fi

For

each

k

such

that

consider

the

sums

of

fundamental

cycles

xk

X

oe

oe

-LRB-

-RRB-

yk

X

oe

oe

-LRB-

-RRB-

where

oe

-LRB-

o

-RRB-

is

the

image

of

the

partition

o

under

the

permutation

oe

and

Ti

is

the

subgroup

of

which

fixes

the

first

i

letters

point-wise

Here

we

identify

Si

with

the

subgroup

of

which

fixes

the

i

largest

letters

point-wise

The

stabilisers

of

xk

and

yk

are

respectively

Xk

and

Yk

if

If

then

the

stabiliser

is

Zk

Also

the

elements

foe

-LRB-

xk

-RRB-

ae

-LRB-

yk

-RRB-

g

where

oe

and

ae

range

respectively

over

the

coset

representatives

of

Xk

and

Yk

are

all

independent

When

n

is

even

and

n

similarly

the

collection

fae

-LRB-

xk

-RRB-

g

for

ae

ranging

over

the

coset

representatives

of

Xk

is

independent

of

all

the

others

A

more

detailed

analysis

shows

that

in

this

case

also

the

set

fae

-LRB-

xk

-RRB-

g

is

itself

independent

Hence

for

xk

-LRB-

respectively

yk

-RRB-

generates

the

transitive

permutation

module

with

stabiliser

subgroup

Xk

-LRB-

respectively

Yk

-RRB-

A

similar

statement

holds

for

the

case

Remark

Note

that

the

Betti

number

of

the

poset

-LRB-

k

-RRB-

can

be

computed

using

the

defining

recurrence

for

the

M?obius

function

-LRB-

see

also

-LSB-

Su

Proposition

-RSB-

-RRB-

We

then

have

-LRB-

-RRB-

k

-LRB-

-LRB-

k

-RRB-

-RRB-

k

X

i

-LRB-

-RRB-

i

-LRB-

-RRB-

-LRB-

-RRB-

k

where

is

the

Betti

number

of

the

poset

The

proof

of

Theorem

shows

that

the

representation

H

-LRB-

-LRB-

k

-RRB-

-RRB-

is

in

fact

determined

by

the

degree

term

in

the

polynomial

-LRB-

He

-RRB-

which

was

shown

to

have

integral

coefficients

Hence

Conjecture

is

equivalent

to

the

conjecture

that

this

polynomial

has

nonnegative

integral

coefficients

From

the

proof

of

Theorem

we

extract

the

following

recursive

algorithm

for

computing

the

homology

representations

Algorithm

Define

polynomials

-LRB-

x

y

-RRB-

n

recursively

by

setting

-LRB-

i

-RRB-

-LRB-

x

y

-RRB-

ty

-LRB-

ii

-RRB-

-LRB-

x

y

-RRB-

-LRB-

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

-RRB-

-LRB-

iii

-RRB-

-LRB-

x

y

-RRB-

coefficient

of

in

-LRB-

x

y

-RRB-

coefficient

of

in

-LRB-

x

y

-RRB-

-LRB-

iv

-RRB-

-LRB-

x

y

-RRB-

-LRB-

x

-LRB-

ty

-RRB-

tx

y

-RRB-

-LRB-

x

y

-RRB-

Then

-LRB-

a

-RRB-

The

characteristic

of

the

top

homology

of

is

given

by

-LRB-

-RRB-

-LRB-

b

-RRB-

The

characteristic

of

the

top

homology

of

-LRB-

k

-RRB-

-LRB-

respectively

-LRB-

k

-RRB-

-RRB-

for

k

n

is

given

by

substituting

x

He

y

Ho

in

-LRB-

x

y

-RRB-

-LRB-

respectively

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

-RRB-

and

then

extracting

those

terms

which

are

homogeneous

polynomials

of

degree

-LRB-

respectively

-RRB-

in

the

fhigi

We

may

now

summarise

the

content

of

the

preceding

conjectures

by

the

statement

that

-LRB-

x

y

-RRB-

is

a

polynomial

in

t

x

and

y

all

of

whose

coefficients

are

nonnegative

integers

Conjecture

implies

that

the

Whitney

homology

-LRB-

see

-LSB-

-RSB-

-LSB-

-RSB-

and

-LSB-

Su

-RSB-

-RRB-

of

the

dual

poset

is

also

a

permutation

module

However

computations

show

that

the

Whitney

homology

of

is

not

in

general

a

permutation

module

The

first

few

polynomials

-LRB-

x

y

-RRB-

are

recorded

below

-LRB-

The

coefficient

of

is

:-RRB-

-LRB-

x

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

For

odd

m

the

representations

Rm

do

not

seem

to

have

especially

nice

properties

We

have

using

Algorithm

Using

the

techniques

in

-LSB-

Su

Section

-RSB-

we

obtain

from

the

defining

equation

of

Theoerm

a

relation

between

the

multiplicities

of

-LRB-

Schur

functions

indexed

by

-RRB-

hooks

in

the

and

the

multiplicity

of

the

trivial

representation

in

Proposition

-LRB-

i

-RRB-

H

-LRB-

em

-RRB-

is

not

a

permutation

module

for

Sm

when

m

is

odd

-LRB-

ii

-RRB-

Let

En

-LRB-

n

-RRB-

denote

the

multiplicity

of

the

trivial

representation

in

H

-LRB-

-RRB-

-LRB-

cf.

Corollary

-RRB-

and

let

ci

-LRB-

j

-RRB-

be

the

multiplicity

in

H

-LRB-

-RRB-

of

the

irreducible

indexed

by

the

hook

-LRB-

j

-RRB-

for

i

such

that

n

Then

En

-LRB-

n

-RRB-

n

X

i

n

-LRB-

-RRB-

i

n

-LRB-

ci

-LRB-

-RRB-

ci

-LRB-

-RRB-

-RRB-

Proof

-LRB-

i

-RRB-

From

Algorithm

we

see

by

induction

that

the

polynomial

-LRB-

x

y

-RRB-

-LRB-

x

y

-RRB-

contains

the

term

-LRB-

-RRB-

n

-LRB-

ty

-RRB-

This

contributes

-LRB-

-RRB-

n

-LRB-

Pn

i

-RRB-

to

the

expansion

of

in

the

homogeneous

symmetric

functions

It

follows

that

if

n

is

odd

the

character

value

of

is

-LRB-

-RRB-

on

a

-LRB-

-RRB-

cycle

while

if

n

is

even

the

character

value

is

-LRB-

-RRB-

on

a

-LRB-

-RRB-

cycle

Consequently

the

homology

of

is

never

a

permutation

module

for

n

-LRB-

ii

-RRB-

Use

-LSB-

Su

Lemma

-RSB-

and

the

fact

that

the

plethystic

inverse

of

P

i

hi

is

P

i

-LRB-

-RRB-

i

where

ssn

is

the

characteristic

of

the

representation

of

Sn

on

the

top

homology

of

the

full

partition

lattice

n

-LRB-

See

for

example

-LSB-

Su

Example

-LRB-

i

-RRB-

-RSB-

-RRB-

Note

that

the

truth

of

our

conjectures

would

imply

that

the

homology

of

the

subposet

-LRB-

k

-RRB-

is

a

permutation

module

for

for

all

k

n

In

particular

the

characteristic

of

the

restriction

of

the

homology

of

to

may

be

easily

computed

from

the

polynomial

expansion

of

-LRB-

-RRB-

given

by

Theorem

We

have

Corollary

As

an

the

homology

of

decomposes

into

a

-LRB-

possibly

virtual

-RRB-

direct

sum

of

induced

modules

as

follows

H

-LRB-

-RRB-

n

X

i

di

-LRB-

n

-RRB-

The

integers

di

-LRB-

n

-RRB-

are

determined

by

the

equations

di

-LRB-

n

-RRB-

-LRB-

n

i

-RRB-

bi

-LRB-

n

-RRB-

-LRB-

i

-RRB-

bi

-LRB-

n

-RRB-

n

Conjecture

implies

that

the

integers

di

-LRB-

n

-RRB-

are

nonnegative

for

all

i

n

-LRB-

From

Corollary

-LRB-

n

-RRB-

and

-LRB-

n

-RRB-

-LRB-

n

-RRB-

-LRB-

n

-RRB-

for

all

n

:-RRB-

We

conclude

this

section

with

some

remarks

on

other

methods

for

showing

that

a

homology

module

is

a

permutation

module

As

argued

in

the

introduction

and

as

shown

by

the

basis

for

-LRB-

-RRB-

constructed

in

Remark

it

is

not

true

that

the

homology

spheres

themselves

are

permuted

by

On

the

other

hand

since

-LRB-

-RRB-

does

contain

a

unique

element

fixed

by

namely

the

two-block

partition

-LRB-

-RRB-

in

which

is

a

singleton

there

is

some

hope

that

the

homology

spheres

of

the

subposets

-LRB-

k

-RRB-

k

n

-LRB-

which

in

fact

contain

a

maximal

chain

fixed

by

-RRB-

are

permuted

by

the

action

of

the

subgroup

While

it

seems

unusual

for

homology

modules

to

be

permutation

modules

for

the

action

of

the

full

automorphism

group

there

are

several

examples

of

this

phenomenon

when

the

automorphism

group

is

restricted

to

a

subgroup

The

examples

we

have

in

mind

arise

as

subposets

of

the

partition

lattice

n

considered

as

modules

for

the

subgroup

Sn

It

was

shown

in

-LSB-

Su

-RSB-

that

the

action

of

Sn

on

the

homology

of

the

subposets

n

-LRB-

k

-RRB-

obtained

by

taking

the

top

k

ranks

of

n

excluding

the

top

element

of

rank

n

is

a

permutation

module

for

Sn

-LRB-

see

-LSB-

Su

Theorem

-LRB-

-RRB-

-RSB-

for

an

explicit

determination

of

the

orbits

-RRB-

Here

k

n

-LRB-

For

k

n

one

obtains

the

regular

representation

of

Sn

a

result

first

obtained

in

-LSB-

-RSB-

-RRB-

The

proof

given

in

-LSB-

Su

-RSB-

for

arbitrary

k

is

by

formal

manipulation

of

the

Frobenius

characteristics

Wachs

-LRB-

-LSB-

-RSB-

-RRB-

has

constructed

bases

of

fundamental

cycles

for

these

subposets

which

are

indeed

permuted

by

Sn

Enumerative

Implications

The

results

of

the

preceding

section

have

consequences

which

may

be

described

in

a

purely

enumerative

setting

By

combining

Theorems

and

we

see

that

our

conjecture

would

have

the

following

intriguing

consequence

denote

as

before

by

the

number

of

alternating

-LRB-

up-down

-RRB-

permutations

on

-LRB-

-RRB-

letters

Conjecture

-LRB-

i

-RRB-

The

integers

fbi

-LRB-

n

-RRB-

i

ng

-LRB-

n

-RRB-

n

-LRB-

bi

-LRB-

n

-RRB-

unless

i

n

-RRB-

defined

by

the

recurrence

bi

-LRB-

n

-RRB-

X

k

k

k

X

r

-LRB-

-RRB-

r

i

k

i

bi?k

-LRB-

n

r

-RRB-

are

nonnegative

-LRB-

ii

-RRB-

The

integers

fEi

-LRB-

n

-RRB-

i

ng

defined

by

the

transformation

Ek

-LRB-

n

-RRB-

Pni

bi

-LRB-

n

-RRB-

n?i

k?i

k

n

are

also

nonnegative

Since

furthermore

we

have

n

X

i

bi

-LRB-

n

-RRB-

i

n

X

i

Ei

-LRB-

n

-RRB-

this

would

yield

apparently

new

refinements

of

the

numbers

It

is

difficult

to

see

how

to

make

an

inductive

argument

in

favour

of

the

positivity

of

the

bi

-LRB-

n

-RRB-

from

the

above

recurrence

Notice

that

the

decomposition

of

Conjecture

is

strongly

reminiscent

of

the

orbit

decomposition

-LRB-

Theorem

-RRB-

for

the

action

of

on

the

chains

of

the

characteristic

of

this

action

is

n

X

i

ai

-LRB-

-RRB-

where

the

ai

-LRB-

-RRB-

count

the

number

of

simsun

permutations

in

with

exactly

i

descents

-LRB-

see

Section

-RRB-

From

Proposition

we

obtain

an

equivalent

implication

of

Conjecture

which

is

perhaps

more

natural

from

the

enumerative

point

of

view

Conjecture

The

number

of

simsun

permutations

in

with

exactly

-LRB-

n

-RRB-

descents

-LRB-

and

hence

the

number

of

Andr?e

permutations

in

with

n

peaks

-RRB-

admits

the

further

refinements

-LRB-

into

nonnegative

integers

-RRB-

an

-LRB-

-RRB-

n

X

i

bi

-LRB-

n

-RRB-

i

n

X

i

Ei

-LRB-

n

-RRB-

In

fact

the

dimension

equality

of

Proposition

strongly

suggests

that

the

homology

representation

of

is

isomorphic

to

an

induced

module

V

where

V

is

a

permutation

module

for

of

dimension

an

-LRB-

-RRB-

whose

orbit

decomposition

is

specified

by

the

numbers

bi

-LRB-

n

-RRB-

-LRB-

cf.

Remark

-RRB-

By

examining

the

dimensions

of

each

induced

module

in

the

decomposition

of

Corollary

for

the

restriction

of

to

we

are

led

to

conjecture

the

existence

of

the

following

apparently

new

refinement

for

the

Genocchi

number

One

way

to

define

the

Genocchi

numbers

is

by

means

of

the

exponential

generating

function

-LSB-

Chapter

Exercise

-LRB-

d

-RRB-

-LRB-

f

-RRB-

-RSB-

X

n

-LRB-

-RRB-

n

-LRB-

-RRB-

x

tanh

-LRB-

x

-RRB-

The

Betti

number

of

is

therefore

related

to

the

Genocchi

number

by

the

equation

-LRB-

-RRB-

Conjecture

The

integers

di

-LRB-

n

-RRB-

of

Corollary

are

positive

and

are

divisible

by

for

all

i

n

Consequently

we

have

the

following

refinement

of

the

nth

Genocchi

number

into

a

sum

of

-LRB-

n

-RRB-

positive

integers

for

n

n

X

i

di

-LRB-

n

-RRB-

For

n

define

polynomials

-LRB-

x

-RRB-

Pn

i

di

-LRB-

n

-RRB-

xi

Then

-LRB-

-RRB-

The

first

few

coefficients

of

the

polynomials

are

as

follows

x

x

x

x

x

Numerical

verification

reveals

that

this

refinement

does

not

coincide

with

any

of

the

refinements

for

the

Genocchi

numbers

discussed

in

the

seemingly

exhaustive

survey

article

of

Viennot

-LSB-

V

-RSB-

All

of

these

phenomena

are

in

need

of

a

combinatorial

explanation

We

have

been

able

to

find

a

refinement

of

the

Genocchi

numbers

into

sums

of

powers

of

by

defining

an

equivalence

relation

on

a

certain

combinatorial

model

for

objects

counted

by

the

Genocchi

numbers

The

combinatorial

interpretation

that

we

use

is

due

to

Viennot

A

more

recent

reference

is

-LSB-

Chapter

Exercise

-LRB-

f

-RRB-

-LRB-

ii

-RRB-

-RSB-

Proposition

-LSB-

V

Proposition

-RSB-

The

Genocchi

number

is

the

number

of

permutations

oe

in

with

the

property

that

oe

-LRB-

i

-RRB-

oe

-LRB-

i

-RRB-

iff

oe

-LRB-

i

-RRB-

is

even

for

i

Now

consider

the

representation

due

to

Foata

and

Sch?utzenberger

of

a

permutation

as

an

increasing

binary

tree

-LRB-

see

-LSB-

FSch

-RSB-

-LSB-

V

-RSB-

and

-LSB-

-RSB-

-RRB-

Then

oe

-LRB-

i

-RRB-

oe

-LRB-

i

-RRB-

iff

oe

-LRB-

i

-RRB-

is

a

right

child

of

oe

-LRB-

i

-RRB-

Thus

A

refinement

of

an

-LRB-

-RRB-

into

sums

of

powers

of

different

from

the

one

predicted

by

Conjecture

has

recently

been

discovered

by

Wachs

-LRB-

-LSB-

-RSB-

-RRB-

who

defines

an

action

of

Sn

on

a

subset

of

the

symmetric

group

which

is

equinumerous

with

the

set

of

simsun

permutations

counted

by

an

-LRB-

-RRB-

is

also

the

number

of

increasing

binary

trees

on

-LRB-

-RRB-

nodes

with

the

property

that

a

node

has

a

right

child

if

and

only

if

it

is

even

We

shall

refer

to

such

a

tree

as

a

Genocchi

tree

We

define

the

following

operation

on

a

Genocchi

tree

choose

a

node

of

degree

and

exhange

the

left

and

right

subtrees

of

this

node

Note

that

a

node

of

degree

necessarily

has

one

odd

child

and

one

even

child

and

is

itself

even

Call

two

Genocchi

trees

equivalent

if

one

can

be

obtained

from

another

by

a

series

of

such

reflections

Clearly

this

defines

an

equivalence

relation

on

the

set

of

Genocchi

trees

whose

classes

are

of

size

where

i

is

the

number

of

nodes

of

degree

in

any

one

representative

of

the

class

It

is

easy

to

see

that

there

is

a

unique

class

of

size

corresponding

to

the

tree

in

which

all

nodes

except

have

degree

Also

the

maximum

number

of

nodes

of

degree

is

-LRB-

n

-RRB-

and

is

attained

when

all

the

even

nodes

with

the

exception

of

-LRB-

-RRB-

have

degree

In

this

case

working

backwards

from

the

node

it

is

clear

that

the

children

of

must

be

and

for

i

n

and

hence

the

representative

tree

for

a

class

of

size

is

uniquely

determined

Proposition

Let

gi

-LRB-

n

-RRB-

denote

the

number

of

equivalence

classes

of

size

under

the

above

equivalence

relation

The

enumerative

refinement

of

the

Genocchi

number

arising

from

this

relation

is

thus

n

X

i

gi

-LRB-

n

-RRB-

where

-LRB-

n

-RRB-

gn

-LRB-

n

-RRB-

for

all

n

For

example

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Partitions

with

number

of

blocks

divisible

by

d

Denote

by

Pm

;d

the

rank-selected

subposet

of

m

consisting

of

all

partitions

such

that

the

number

of

blocks

is

divisible

by

d

-LRB-

In

the

preceding

sections

we

wrote

for

the

poset

:-RRB-

It

is

natural

to

wonder

whether

these

posets

yield

interesting

homology

representations

for

d

Again

the

representations

may

be

computed

from

the

plethystic

generating

functions

of

-LSB-

Su

Theorem

-RSB-

It

is

easy

to

see

that

these

representations

are

not

in

general

permutation

modules

for

instance

by

examining

the

simplest

case

of

partitions

of

consisting

of

d

blocks

The

only

nontrivial

partition

left

invariant

under

the

action

of

the

-LRB-

-RRB-

cycle

which

fixes

is

the

two-block

partition

with

forming

a

block

by

itself

Hence

for

d

the

trace

of

a

-LRB-

-RRB-

cycle

on

the

chains

is

and

consequently

on

the

reduced

homology

it

is

-LRB-

-RRB-

Let

m

;d

denote

the

absolute

value

of

the

M?obius

number

of

Pm

;d

-LRB-

In

particular

:-RRB-

As

in

the

proof

of

Proposition

we

obtain

the

following

generating

function

for

the

Betti

numbers

m

;d

Proposition

-LRB-

ex

-RRB-

d

X

i

X

n

-LRB-

-RRB-

n

nd

i

;d

xnd

i

-LRB-

nd

i

-RRB-

X

n

-LRB-

-RRB-

n

nd

;d

-LRB-

ex

-RRB-

nd

-LRB-

nd

-RRB-

Denote

by

dn

the

d-divisible

lattice

first

studied

in

-LSB-

CHR

-RSB-

that

is

the

sublattice

of

nd

consisting

of

partitions

whose

block

sizes

are

divisible

by

d

Remark

-LRB-

iv

-RRB-

points

out

the

relation

between

the

Betti

numbers

of

and

We

do

not

know

if

there

is

any

relation

between

the

Betti

number

nd

;d

of

the

poset

Pnd

;d

and

the

Betti

number

of

the

poset

dn

There

is

some

enumerative

motivation

for

studying

these

posets

Consider

the

case

of

Pnd

;d

Define

an

-LRB-

n

-RRB-

by

-LRB-

n

-RRB-

matrix

Mn

by

letting

the

element

in

row

i

and

column

j

i

j

n

be

the

Stirling

number

of

the

second

kind

S

-LRB-

di

dj

-RRB-

if

i

j

define

the

-LRB-

i

-RRB-

entry

to

be

for

all

i

and

the

-LRB-

j

-RRB-

entry

to

be

zero

for

all

j

The

matrix

Mn

is

lower

triangular

with

's

on

the

diagonal

hence

its

inverse

is

a

lower

triangular

integer

matrix

with

's

on

the

diagonal

Let

bi

j

denote

the

-LRB-

i

j

-RRB-

entry

of

the

inverse

From

the

recursive

definition

of

the

M?obius

function

it

follows

that

bi

-LRB-

-RRB-

i

-LRB-

i

-RRB-

d

;d

-LRB-

Remark

-LRB-

i

-RRB-

is

a

special

case

of

this

observation

-RRB-

More

generally

we

have

that

bi

j

is

the

Whitney

number

wi

i?j

of

the

first

kind

of

the

poset

Pnd

;d

See

-LSB-

Ai

Proposition

-RSB-

-LRB-

The

Whitney

numbers

of

the

second

kind

are

the

Stirling

numbers

S

-LRB-

di

dj

-RRB-

:-RRB-

Here

wi

k

P

;d

rank

-LRB-

x

-RRB-

i?k

-LRB-

x

-RRB-

P

;d

rank

-LRB-

x

-RRB-

i?k

-LRB-

-RRB-

i?k

-LRB-

H

-LRB-

x

-RRB-

Pid

;d

-RRB-

where

denotes

the

Betti

number

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-LSB-

Ai

-RSB-

M.

Aigner

Combinatorial

Theory

Grundlehren

der

mathematischen

Wissenschaften

SpringerVerlag

-LRB-

-RRB-

-LSB-

-RSB-

K.

Baclawski

Whitney

numbers

of

geometric

lattices

Adv.

in

Math

-LRB-

-RRB-

-LSB-

-RSB-

K.

Baclawski

Cohen-Macaulay

ordered

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Algebra

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-RRB-

-LSB-

-RSB-

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Bj?orner

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partially

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-LSB-

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-LRB-

-RRB-

-LSB-

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-RSB-

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size

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symmetric

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-RRB-

-LRB-

-RRB-

-LSB-

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Euler

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Accademia

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Garsia

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-RRB-

-LSB-

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-RRB-

-LSB-

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J.

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-RRB-

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Solomon

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Wachs

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communication

-LRB-

-RRB-

andre permutations
homology representation
permutation module
simsun permutations