n Color partitions with weighted differences equal to minus two

VOL. 20 NO. 4 (1997) 759-768

n-COLOR PARTITIONS WITH WEIGHTED DIFFERENCES

EQUAL

TO MINUS

TWO

A.K. AGARWAL

Mathematical Sciences Division InstituteofAdvanced

Study

in ScienceandTechnology

Jawahar

Nagar,

Khanapara

Guwahati 781022,

INDIA

R. BALASUBRANANIAN

The InstituteofMathematical Sciences

C T.

Campus

Madras- 600113,

INDIA

(ReceivedNovember6,

1995)

ABSTRACT.

In

this

paper

we

study

those n-color partitions of AgarwalandAndrews, 1987,in which each pair ofparts has weighted difference equal to 2 Results obtained in this

paper

for these

partitions include several combinatorialidentities,recurrencerelations, generating functions, relationships

withthe divisor function andcomputer producedtables.

By

usingthesepartitionsanexplicit expression

for the sumofthe divisorsofoddintegersisgiven Itisshownhow these partitionsarise inthestudyof

conjugate and self-conjugate n-color partitions

A

combinatorial identity for self-conjugate n-color partitions is also obtained. We conclude by posing several open problemsinthe last section

KEY WORDS AND PHRASES:

Partitions, combinatorialidentities,recurrencerelations, generating

functions.

1991

AMS

SUBJECT

CLASSlICATION

CODES:

Primary: 05A15, 05A17, 05A19, Secondary 11B37, 11P81

1.

INTRODUCTION, DEFINITIONS AND NOTATIONS

n-color partitions

(also

called partitionswith

"n

copies of

n")

were imroduced by Agarwal and

Andrewsin

[2].

Theseare thepartitionsin which a partofsizen,cancomeinndifferentcolorsdenoted

by subscripts nl,n2, n,. Thus, for example,then-color partitions of3are

31

32

33

2111

211

111111

If

P(v)

denotethenumberof n-color partitions of v,then it wasshownin

[2]

that

1

+

E

P()q

H

(I

q)-.

(I

I)

v=l n=l

Itwaspointedoutin

[2]

that sincethefight handsideof

(l.l)

isalsoageneratingfunctionfor the

MacMahon’splane partitionfunction sothe numberofn-color partitionsofvequalsthenumber ofplane partitions ofv

In

termsof n-color partitionsaclassofnewRogers-Rarnanujamtype identities wasgiven

in the same

paper.

Further Rogers-Karnanujam

Type

identitiesusing n-color partitions were foundin Thiswas oneofthe

advantages

of

studying

n-color partitions asthere are noRoers-Karnanujam

Type

identitiesfor plane partitions

In

thispaperwedefineconjugate and self-conjugate n-color partitionsand obtain various combinatorial identities using these definitions. This will give another

advantage

of

760 A K AGARWAL AND R BALASUBRANANIAN

herethatconjugateandself-conjugate

d(n)-color

partitions,where

d(n)

isthe numerof positivedivisors ofnhave been studiedby Agarwaland Mullen in

[3]

Wenowgivesomedefinitions whichweshall use in thispaper

DEFINITION

1. Let

H

_(al)b:

+

(a2)b2

+

+

(a)b

be an n-colorpartition of We call

(a,),_b,+l

the conjugate of

_(a,)b

An

n-colorpartition of obtained from 17 by replacingeach of its partsbyitsconjugatewillbecalledtheconjugate of

H

and will bedenoted by lI For example,if we consider

YI

₅₂

₊

3,ann-color partition of8, then l’I

55-2+1

+

33-+

54

+

33

DEFINITION

2.

We

shallcall ann-color partitionto beself-conjugateif it is identical with its

conjugate

I’V

Thus

53

+

32

+

1

is aself-conjugaten-colorpartition of9.

DEFINITION 3

(see [2]).

The weighted difference of any pair of parts

m,,n

is defined by

m-i-n-j.

Throughoutthispaper

A()

willdenote thenumberof n-color partitions of positive integeruwhere

theweighteddifferenceofeachpair of partsis 2 Thus

A(,)

0 if

_<

0.

A(m, )

willdenote the numberof partitionsof enumeratedby

A()

withtheadded restriction that there beexactlymparts

Obviously,

A(m, )

0 for

<

m

Weshall also usenearestintegerfunctions Wedefine asfollows

[zj

r,

(1.2)

whoreristheunique integer suchthat

1 1

x---<r<x-F-.

2- 2

2.

COMBINATORIAL IDENTITIES

In

this section we shall prove several combinatorial identities

Our

first identity is an easy consequence ofthe definitionof conjugacy

TItEOREM

2.1. Let

B(,)

denote the numberof n-color partitions ofusuchthat ineach pair of partsm,,

n3(m

> n)

nisthe arithmetic meanof thesubscripts and j. Then

A(u)

B(u),

EXAMPLE.

A(5)

11 The relevantpartitionsin this case are 51,52,53, 54,55,4411,3122, 321, 331111,

2211

llll,

1111111111.

B(5)

is also equalto 11, since in this case the relevantpartitionsare 51,52,53,54,5s, 4111,3321,

3222,31111, 2111111,

1111111111

PROOF.

Conjugacy is the natural bijection between the twoclasses.

To

see this, let 17 bca partition enumerated by

A(v).

Thatis,eachpair of partsm,_n3in

H

satisfies the condition

m-n-i-j= -2.

(21)

Weclaimthateachpair of parts

pq,

r in

II

satisfies the condition

We

seethat

2

:

Pp-q+l,rr-s+l 1"I

=, _p-

_(p-

q

+

l)

r

(r

s

₊

l)

-2, (by

(2.1))

which is the same as

(2.2).

To

see the reverseimplicationletabe apartitionenumeratedby

B(v)

That is, eachpair ofparts

pq,

r

E a satisfies

(2.2)

Wewantto

prove

thateach pair of partsm,,

_n

incr" satisfies

(2

1)

761

)’)’)n E(yc

_::

PrI,_{m_+l)nn_2+} (O"

(m-

i-+-

1)

/

(n-

3/

1)

which is the same as 2.1 This

completes,,the,

lro,o,f

ofTheorern2

To

illustratethebijection ofTheorem2.1,weprovidetheexample forv 6

Partitionsenumerated

by

A(6)

61

62

63

64

65

66

5511

4222

4321

441111

3131

33111111

212121

2211111111

111111111111

Conjugacy Partitionsenumerated

by/3(6)

66

65

63

61

5111

4321

4222

411111

3333

31111111

222222

2111111111

111111111111

REMARK.

Using the idea of conjugacy, partition identities have been obtained for ordinary partitions

[4,

Theorem 343, p.

274]

andfor

d(n)-color

partitions

[3,

Theorem4 1,p

128]

Following

themethodof

proof

ofTheorem 2 1, one canproveitsfollowing generalization

TIEOREM

2.2. Let

Ak(v)

denotethe number of n-color partitions ofv suchthat eachpairof parts has aweighted differenceequaltok 2 Let

Bk(v)

denote the number of n-colorpartitionsof such that eachpair ofparts m,,

_n

satisfies the condition

+

j 2n k Then

A

()

B

k

_(v)

REMARK. Theorem 2.1 is aparticularcase k 0ofTheorem 2.2.

Theorem 2 is acombinatorialidentitybetweentwon-color partitionfunctions

Our

nexttheorem isacombinatorial identity betweenann-color partition functionon one sideand anordinary partition

function on theother side

TItEOREM

2.3. Let

C(m, v)

denote the numberofordinary

_partitio.ns

ofallnumbers

<

vwith minimum partmand the differencesbetween parts 0 orm 1 Then

A(m, v)

C(m, ).

(2

3)

EXAMPLE. Considerthe case in whichv 11 andrn 3 Wesee that

A(3, 11)

5,since the relevant partitions are 991111,

762121,

643122, 533131, 414132; also

C(3,11)=5;

in this casethe relevantpartitionsare: 3,3

₊

3, 3

₊

5,3

₊

3

₊

3,3

₊

3

₊

5.

PROOF.

Forrn 1, thetheorem isobviouslytruesince

A(1, v)

v, the relevant partitionsare

1, v2,

v

also

C(1, v)

v,in this case the relevantpartitionsare

1, 1+1, 1+1+1

1/1+...+1.

Nowwe considerthe case when rn

>

2.

In

this case weshallfirstprove that

762 AKAGARWALAND R BALASUBRANANIAN

To prove

(2 4),

wesplitthepartitionsenumeratedby

A(m, v)

into threeclasses

(i)those that donotcontain

kk(k >_ 1)

asapart,

(ii)those that contain

11

asa

pan,

and

(iii)

thosethat containkk,

(k > 1)

as apartbutnot

₁₁

We

now transform thepartitions in class

(i)

by deleting 1fromeach

pan

ignoring the subscripts Obviously, the transformed partition still satisfies the weighted difference condition so it will be a

partitionenumeratedby

A(m,v

m).

In

class

(ii)

weobserve that

11

can

appear

onlywith

kk(k >_ 1)

in ordertosatisfytheweighteddifference condition. Also,notwo

pans

kk

and

It

withk

>

1 and

>

Ican

appear together Thisimplies that correspondingtoeach valueof

rn(m > 1),

thereisone andonlyone

partitionin class

(ii)

Finally, we transformthe partitionsin class (iii) by replacing

kk

by

(k

+

and then subtracting 2 from each part This will give apartition ofu-2m

+

1 into m parts with

(k-1)k_

as a

pan.

Therefore, the actual number of panitions which

belong

to class

(iii)

is

A(m,

-

2m

₊

1)- A(m,

,-3m

+

1),

where

A(m,

-

3m

+

1)

is the number of panitions of v-2m

₊

1 into m

pans

whichare free from the

pans

like

kk(k > 1).

The above transformations

clearly prove

(2 4)

Now

we set

Then

Next,

weset

Then

Hence

D(m,v)

A(m,v)

A(rn,

v-

m),

Vv.

(25)

l"

0 if u

<

m

D(rn,,t,,)

_{D(m,t,,_}

_2m

+

l)

+

l, if_>m

(26)

E(rn, v)

D(m, v)

D(rn,

v 2m

+

1),

Vv.

(2 7)

0 if

,

<

m

(2.8)

E(m,

1

1 if t,>m

qm

1--q

Usingthe extreme oftheforegoing string of equations in

(2

7)

and then theresulting equation in

(2 5),

weget

(2 9)

comparingthe coefficientsof

q",

weget

76 3 NOTE. Weremark here that the first part of

(2.9)

can, alternatively, beproved by using

(2 4)

Since if

fm

(q)

]

A(m,

v)q"

then

(2 4)

yields

v--2

fro(q)

qm

fm(q)

q-

_qm-l

_fm(q)

_q3m-l

_fm(q)

_H-

_qm(1

_q)-I

GENERATING FUNCTIONS

Our

discussionintheprecedingsectionleads ustothefollowing TItEOREM3.1.

(i)

’

A(I’

v)q

(1

q)2

=1

(ii)

E

A(m’v)qV=

qm

v=2

(1

q)(1

qm)(1

q2m-1)’

rn

>_

2

(iii)

A(m)q

v--1

qm

q

+

(1-q)2

(1-q)(1--qm)(1-q2m-1)

4. AN

EXPLICIT

FORMULA

In

this section wegivenanexplicit formulafor

A(m, v)

We provethefollowing

THEOREM

4.1.

Let

rbe thequotient of and denotes thenearestintegerfunction definedby

(1

2)

above Then

{r

1 if m

>_

via

W---A(m, v)

if m

<

v+l 2m-1

2=0

v+l

PROOF.

Ifm

>

_---,

thenclearly

A(m,v)

1since theonlyrelevantpartitionis

(v

m

₊

_{1)v_m+}

m-1 parts

v+l

sowe considerthe case whenm

<

_-T--

In (2.6)

abovewe write v

k(2m- 1)

+g.

Then

D(m,

k(2m

1)

+

e)

D(m,

(k

1)(2m

1)

+

g)

+

1

D(m,

(k

2)(2m

1)

+

g)

+

2

D(m,

g)

+

k

={k+l

if

e>m

k ifg<m

Hence

(4

1)

(2.5)

in viewof

(4.1)

canbe written as

0 if u<m

A(m,v)=

A(m,-m)+[

2-_

if

v_>m

764 A KAGARWALAND R BALASUBRANANIAN

A(m,

rm

₊

s)

a(m,

(r-1)m

+

s)

_+]

-t-)r -I-9-)

r-1

2rn-1

1=0 ThisprovesTheorem 4

5.

A RELATIONSHIP WITH DIVISOR FUNCTION

In

this section weshall provearelationship between

A(v)

and

d(v)

where

d(v)

denotes the number

of positivedivisorsofv

THEOREM

5.1. Letasequence

{b}

bedefinedby

bl

1,

b

d(2v-

1)-d(

i)

1,v

_>

2

Let

B

bethesequence of the partialsumsof

b

Then

b=A(v)-2A(v-1)+A(v-2),

v>_l

(5 1)

hence,

B.

A(v)-

A(v-1),

v>l. (52)

PROOF.

From

Theorem 3

(iii),

wehave

q2

(1

-q)

q2

Now writingthe even

m’s

andodd

m’s

separatelyin

c(q),

weget

q21+

q2t

c(q)

t>

l

q2e

+

e>

1- q2e-1

q2+1

E

f

:

-q-2e

+

b(q)

e(q)

+

b(q),

say

q2t

q-le(q)

1

q2e"

Replacingqby

V/’,

we get

765 Hence

Thus

Now

"(q)

=E

d(rt’)q2n+l

--E

d(m-1)

qm"

n>_l m>_l 2

m>_l 2

qm

E

q(2m-1)(k-1)

m=l 1

q2m-1

m=l k>_l

E

q2mk-m-k+l

Hence

Hence

q-la(q2)

E

q4mk-2m-2k-4-1__

E

q(2m-1)(2k-1)

m,k m>_l,k>l

-

d(n)qn

E

d(2rn-1)q

2m-1

odd m>l

a(q2)

E

d(2m-

l)q2m

a(q)

E

d(2m-

l)q

m.

m>_l

Putting these resultstogether,we get,

(l

q)2

E

A(v)q

-

d(2rn

1)q,

E

d(rn

-1)q,

rn>_l 2

E

(d(2m-1)-d(m-1))qm-Eqm.

_{m>_2}

q2

(l-q)

p

_b

B

_A()

1 1 1 1

2 1 2 3

3 0 2 5

4 1 3 8

5 0 3 11

6 1 4 15

7 1 3 18

8 3 6 24

9 2 4 28

10 1 5 33

11 1 6 39

12 1 7 46

A(v)

Theprocedureis illustrated in thefollowingtable

On

comparingthe coefficientsof

q"

inthe last equation, we areledto

(5

1)

(5 2)

follows from

(5

1) immediatelyonce wenotethat thefight-handsideof

(5.2)

isthe nthpartial

um

oftheright-handsideof

(5.1)

Thiscompletestheproof ofTheorem 5 1.

766 A K AGARWAL AND R BALASUBRANANIAN

Fromthetable it is clear that each

A(,)

equalsthe sumofthe numberimmediatelyabove it in the table and the number in the same row in the preceding column.

6.

n-COLOR

PARTITIONS

WITH WEIGHTED DIFFERENCES :2AND

kk (k _> 1)

AS

A PART.

It iseasily seen that

D(m, ,)

definedby

(2 5)above

isthe numberofthosepartitions which are enumerated by

A(m,v,)

and have

kk(k >_

1)

as apart Let

D(,)

equal

D(m,,)

Following the methodsof precedingsections we caneasilyprovethe following results:

THEOREM

6.1. Let

F(m,

,)

denote the numberof ordinary partitions ofall numers

_<

,

into parts where the lowest part ismwhichdoesnotrepeat and the differences between parts are 0 orm 1 Then

D(m, ,)

V(m, ,).

EXAMPLE.

Considerthe case when rn 2 and v 11.

We

see that

D(2, 11)

4, since the relevant partitions are 101011, 9r22, 8433, 7144, also,

F(2,11)=

4, since in this case the relevant

partitionsare

2,2+3,2+3+3,2+3+3+3.

THEOREM6.2.

q

(i)

D(I’

u)q’

1 q

(ii)

D(m;

,)q’

qm

(1

--q)(l _q2m-l)

(iii)

E

D(’)q’

(1

q)

b’--1

q)(

qm-)"

THEOREM

6.3.

THEOREM6.4. Letkbe thequotient of Then

k-1

A(m, ,)

Z

D(m,

,

mj). 3=1

TItEOREM

6.5.

1

D(,)-

D(,- 1)

_d(2,-

_1)-1

if_if ,=1_,>1"

UsingTheorems 6.3 and 6.5,we shallnowprovethefollowing explicit expression forthe sum of the divisorsofoddnumbers:

THEOREM6.6.

d(2j-1)

[

2---

1"

.7=1

76 7

that is

D(v)

d(2j-

1)-

(v-

1),

3=1

m_>2 3=1 Thisproves Theorem6.6byTheorem 6.3

7. A

COMBINATORIAL

IDENTITY FOR SELF-CONJUGATE r-COLOR PARTITIONS

In

this section we shall prove a combinatorial identity which involves self-conjugate n-color

partitionsdefined in Section above.

TttEOREM

7.1. Let

F(v)

denote thenumberof self-conjugaten-colorpartitionsofvand

G(v)

denotethosen-color partitionsofvwhere each pair of partshasweighteddifference greaterthan and even parts

appear

withevensubscripts and oddwithodd Then

F(v)

G

(v)

forall

EXAMPLE.

F(7)

5; therelevantpartitionsare

74,

53

+

11

+

11,

32

+

G(7)

5,in this case therelevantpartitionsare

PROOF.

We observe thatif ann-oder partitionisself-conjugatethen in each part m,of it,m must beodd

Because,

m

m,_+l

=

m 2i 1. Thusif weignorethesubscripts ofallpartsin a self-conjugate n-color partition of v,wegetauniqueordinarypartition ofvintooddparts.Conversely,if we consider an ordinarypartition ofvintooddpartsandreplaceeach part 2a 1by

(2a

1)a

We

get a

uniqueself-conjugaten-colorpartition ofv. Thisbijectionshows that the numberof self-conjugate n-color partitions ofvequals thenumberof ordinary partitions ofvintooddparts. Thatis,

1

+

F(v)q"

H

1

v=l n=l 1

q2n-1

Nowanappealto thefollowing theorem 1,Theorem 1.4,p.

301]

"Let

R(v)

denote thenumberofn-colorpartitions ofvsuch that each pair ofparts hasweighted

differencegreaterthan 1andevenparts appearwithevensubscriptsandoddwithodd Let

S(v)

denote thenumberof ordinary partitions ofvintodistinct parts. Then

R()

S()"

provesTheorem 7.1.

$.

A TABLE FOR

A(m,

In

this sectionwe give abrief table of

A(m, v)

which was obtained on computer by usingthe recurrence relation

(2 4)

above. The followingobservations can serveasacheckonthetable

Observation

A (1, v)

v,

Observation 2

A(m, v)

1, if m

>

Observation 3

A(m, 2m)

2

Observation 4

A(m,

2m

+

1)

2

We

remark here that Observation was

proved

inthe

proof

of Theorem2 3 whileObservation2in theproof of Theorem4.1 Observations 3and4follow immediately oncewe note that theonly partitions

enumeratedby

A(m, 2m)

are

(rr

+

1)m+i

768 AK. AGARWAL AND R BALASUBRANANIAN and those enumeratedby

A(m,

2m

₊

₁₎

are

4 5 6

7

8 9 10 11 12

13

14 15

(mq-2)m+2

and

_32.

m-1 m-1

1 2 3 4 5 6 7 8 9 10 11

"12

13 14 15 1

2 1

3 1 1

4 2 1 1

5 3 1 1 1

6 4 2 1 1 1

7 5 2 1 1 1 1

8 7 3 2 1 1 1 1

9 8 4 2 1 1 1 1 1

10 10 4 2 2 1 1 1 1 1 11 12 5 3 2 1 1 1 1 1 1

12 14 6 4 2 2 1 1 1 1 1 1

13 16 7 4 2 2 1 1 1 1 1 1 1

14 19 8 4 3 2 2 1 1 1 1 1 1 1

15 21 9 5 4 2 2 1 1 1 1 1 1 1 1

9:

CONCLUSION

The most obviousquestions arising fromthiswork are:

1. Dothe combinatorial identities ofthispaperhave theiranalytic counterparts9 Ourmainhope in

pursuingthe analytic aspects of these results is that we shall find q-identitiesresembling Euler’s

identities

[4,

p

277]

orRogers-Ramanujan identities

[4,

p

290]

which willshedlight onhowto proceedwithfurther studyof the identitiesofthispaper.

2 Isitpossibletoprovethe identitiesofthispaper graphically9

3.

By

usingn-color partitionswehavegivenanexplicit expression for

d(2j-

1)

Isitpossibleto find similarexpressionsfor d(2j)?

3=1

ACKNOWLEDGMENT.

The authorswould like to thank Dr. Venkatest Rarnan for his computer assistance

[1]

AGARWAL, A.K,

Rogers-Ramanujam identities for n-color partitions,

J.

Number

Theory

28

(1958),

299-305

[2]

AGARWAL,

A.K. and

ANDREWS, G.E,

Rogers-Rarnanujam identities for partitions with

"N

copiesofN",

J.

Combin. Theory

Set. A

45, No.

(1957),

40-49

[3]

AGARWAL,

A.K.

and

MULLEN,

G.L.,

Partitions with

"d(a)

copies of

a",

Combin. Theory

Ser.

A,

48

(1) (1988),

120-135.