# n Color partitions with weighted differences equal to minus two

## Full text

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

EQUAL

A.K. AGARWAL

### Study

in ScienceandTechnology

Jawahar

Khanapara

Guwahati 781022,

### INDIA

R. BALASUBRANANIAN

The InstituteofMathematical Sciences

this

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

CLASSlICATION

### CODES:

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

1.

### INTRODUCTION, DEFINITIONS AND NOTATIONS

n-color partitions

### (also

called partitionswith

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

If

### P(v)

denotethenumberof n-color partitions of v,then it wasshownin

that

1

### (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

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

of

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760 A K AGARWAL AND R BALASUBRANANIAN

herethatconjugateandself-conjugate

partitions,where

### d(n)

isthe numerof positivedivisors ofnhave been studiedby Agarwaland Mullen in

### [3]

Wenowgivesomedefinitions whichweshall use in thispaper

1. Let

### (a)b

be an n-colorpartition of We call

the conjugate of

### An

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

### H

and will bedenoted by lI For example,if we consider

### +

3,ann-color partition of8, then l’I

2.

### We

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

conjugate

Thus

### 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

0 if

0.

### A(m, )

willdenote the numberof partitionsof enumeratedby

Obviously,

0 for

### <

m

Weshall also usenearestintegerfunctions Wedefine asfollows

r,

### (1.2)

whoreristheunique integer suchthat

1 1

2- 2

2.

### In

this section we shall prove several combinatorial identities

### Our

first identity is an easy consequence ofthe definitionof conjugacy

2.1. Let

### B(,)

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

### > n)

nisthe arithmetic meanof thesubscripts and j. Then

EXAMPLE.

### A(5)

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

llll,

### B(5)

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

3222,31111, 2111111,

### 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

r in

### II

satisfies the condition

seethat

2

### :

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

=, p-

q

r

s

-2, (by

### (2.1))

which is the same as

### To

see the reverseimplicationletabe apartitionenumeratedby

### B(v)

That is, eachpair ofparts

E a satisfies

Wewantto

### prove

thateach pair of partsm,,

incr" satisfies

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761

)’)’)n E(yc

### ::

PrI,m_+l)nn_2+ (O"

i-+-

/

3/

### 1)

which is the same as 2.1 This

lro,o,f

### To

illustratethebijection ofTheorem2.1,weprovidetheexample forv 6

Partitionsenumerated

by

### 111111111111

Conjugacy Partitionsenumerated

### REMARK.

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

Theorem 343, p.

andfor

partitions

Theorem4 1,p

Following

themethodof

### proof

ofTheorem 2 1, one canproveitsfollowing generalization

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

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

2.3. Let

### C(m, v)

denote the numberofordinary

ofallnumbers

### <

vwith minimum partmand the differencesbetween parts 0 orm 1 Then

### 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,

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

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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

asa

and

### (iii)

thosethat containkk,

as apartbutnot

### 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

class

weobserve that

can

onlywith

### kk(k >_ 1)

in ordertosatisfytheweighteddifference condition. Also,notwo

and

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

by

### +

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

### +

1 into m parts with

as a

### pan.

Therefore, the actual number of panitions which

to class

is

2m

,-3m

where

3m

### 1)

is the number of panitions of v-2m

1 into m

### pans

whichare free from the

like

### kk(k > 1).

The above transformations

clearly prove

we set

Then

weset

Then

v-

Vv.

0 if u

m

2m

l, if_>m

v 2m

Vv.

0 if

m

1

1 if t,>m

### qm

1--q

Usingthe extreme oftheforegoing string of equations in

### 7)

and then theresulting equation in

weget

### (2 9)

comparingthe coefficientsof

### q",

weget

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76 3 NOTE. Weremark here that the first part of

### (2.9)

can, alternatively, beproved by using

Since if

then

yields

v--2

q-

H-

=1

(ii)

v=2

rn

2

v--1

q

4. AN

### In

this section wegivenanexplicit formulafor

### A(m, v)

We provethefollowing

4.1.

### Let

rbe thequotient of and denotes thenearestintegerfunction definedby

above Then

1 if m

if m

v+l 2m-1

2=0

v+l

Ifm

thenclearly

### A(m,v)

1since theonlyrelevantpartitionis

m

### 1)v_m+

m-1 parts

v+l

sowe considerthe case whenm

-T--

abovewe write v

Then

1

2

k

if

k ifg<m

Hence

(4

in viewof

canbe written as

0 if u<m

if

### v_>m

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764 A KAGARWALAND R BALASUBRANANIAN

rm

### +]

-t-)r -I-9-)

r-1

2rn-1

1=0 ThisprovesTheorem 4

5.

### In

this section weshall provearelationship between

and

where

### d(v)

denotes the number

of positivedivisorsofv

### THEOREM

5.1. Letasequence

bedefinedby

1,

1,v

2

Let

### B

bethesequence of the partialsumsof

Then

v>_l

hence,

v>l. (52)

Theorem 3

wehave

### q2

Now writingthe even

andodd

separatelyin

weget

l

1- q2e-1

say

1

Replacingqby

we get

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765 Hence

Thus

Now

n>_l m>_l 2

m>_l 2

m=l 1

m=l k>_l

Hence

Hence

m,k m>_l,k>l

## -

2m-1

odd m>l

l)q2m

l)q

### m.

m>_l

Putting these resultstogether,we get,

rn>_l 2

m>_2

p

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

follows from

### (5

1) immediatelyonce wenotethat thefight-handsideof

isthe nthpartial

### um

oftheright-handsideof

### (5.1)

Thiscompletestheproof ofTheorem 5 1.

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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.

### PARTITIONS

WITH WEIGHTED DIFFERENCES :2AND

### AS

A PART.

It iseasily seen that

definedby

### (2 5)above

isthe numberofthosepartitions which are enumerated by

and have

as apart Let

equal

### D(m,,)

Following the methodsof precedingsections we caneasilyprovethe following results:

6.1. Let

### ,)

denote the numberof ordinary partitions ofall numers

### ,

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

### EXAMPLE.

Considerthe case when rn 2 and v 11.

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)

1 q

(ii)

b’--1

### THEOREM

6.3.

THEOREM6.4. Letkbe thequotient of Then

k-1

mj). 3=1

6.5.

1

### 1)-1

ifif ,=1,>1"

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

THEOREM6.6.

.7=1

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76 7

that is

d(2j-

### 1),

3=1

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

7. A

### In

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

partitionsdefined in Section above.

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

forall

### F(7)

5; therelevantpartitionsare

74,

11,

### G(7)

5,in this case therelevantpartitionsare

### PROOF.

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

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

### We

get a

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

1

1

v=l n=l 1

### q2n-1

Nowanappealto thefollowing theorem 1,Theorem 1.4,p.

### R(v)

denote thenumberofn-colorpartitions ofvsuch that each pair ofparts hasweighted

differencegreaterthan 1andevenparts appearwithevensubscriptsandoddwithodd Let

### S(v)

denote thenumberof ordinary partitions ofvintodistinct parts. Then

### S()"

provesTheorem 7.1.

\$.

### 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

v,

Observation 2

1, if m

Observation 3

2

Observation 4

2m

2

### We

remark here that Observation was

inthe

### proof

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

enumeratedby

are

### 1)m+i

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768 AK. AGARWAL AND R BALASUBRANANIAN and those enumeratedby

2m

are

4 5 6

7

8 9 10 11 12

14 15

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

p

### 277]

orRogers-Ramanujan identities

p

### 290]

which willshedlight onhowto proceedwithfurther studyof the identitiesofthispaper.

2 Isitpossibletoprovethe identitiesofthispaper graphically9

3.

### By

usingn-color partitionswehavegivenanexplicit expression for

### 1)

Isitpossibleto find similarexpressionsfor d(2j)?

3=1

### ACKNOWLEDGMENT.

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

### AGARWAL, A.K,

Rogers-Ramanujam identities for n-color partitions,

Number

28

299-305

A.K. and

### ANDREWS, G.E,

Rogers-Rarnanujam identities for partitions with

copiesofN",

Combin. Theory

45, No.

40-49

and

Partitions with

copies of

Combin. Theory

48

120-135.

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