agarwal a k - an analogue of euler's identity and new rial properties of n-colour composition - pp

8/9/2019 Agarwal a K - An Analogue of Euler's Identity and New rial Properties of N-Colour Composition - pp

1/9

An Analogue of Euler's Identity and New Combinatorial Propertiesof n-Colour CompositionsA.K. AGARWAL1Centre for Advanced Study in MathematicsPanjab University, Chandigarh-160014, India

e-mail: [email protected]

An analogue of Euler's partition identity: "The number of partitionsof a pootive inte~ v into odd parts equals the number of its partitionsinto distinct parts"is obtained for ordered partitions. The ideas developedare then used in obtaining several new combinatorial properti~ of then-rolour compootions introduced recently by the author.

1. introductionThe first partition identity whichstates" The numberof partitions of a positiveinteger nto odd parts equals he numberof its pazotitionsnto distinct part" isdue to Euler [6, p.277]. In Section 2 we shall prove analytically as well as com-bina-toriallyan analogue f Euler's dentity for orderedpartitions (also called

The ideasdevelopedn Section2 will becompositions by P.A. MacMahon)used n Section3 in obtainingseveral ew combinatorialpropertiesof n-colourcompositionsntroduced ecentlyby the author in [1]. First we recall the [0)-lowing definitions which will be used n the sequel.Definition 1 ([3]). An "odd-even"partition of a positive ntegerv is a parti-tion in which the parts (arranged n ascendingorder) alternate in parity startingwith the smallestpart odd. Thus, or example, he "odd-even"partitions of 7are: 7,1+6,3+4.

ar )r '2~--Supportedby csm ~a.rcl1 Gnnt No.25(Ol28)/O2/EMR-ll

1


2/9

r rwhere al > ~ > ... > ar ;?:0,bI > b2 > ... > br ;?:0 and v = r.4- }:= ai + }:= bi

i=l i=lis called a Frobenius artition of v. We note that eachpartition can be repre-sented by a. Frobenius notation since f we delete from the Ferrers graph of thepartition the main diagonalwhich suppose o~sses r dots hen the remainingrowsof dots o the right of the diagonalare enumeratedo provideone strictlydecreasingsequence f r nonnegative ntegers (the rth such row might be emptythus producing0). The remainingdots below he diagonalare enumerated ycolumns o provide a second trictly decreasing equencef r nonnegativen-tegers the rth sud1column might be empty thus producing0). The resultingtwo sequences l'e then presented n the Frobenius notation. For example, the

6 5 2 05 3 2 1robenius notation for 7+7+5+4+4+1 isDefinition 8. ([2]). An n-colourpartition (or a. partition with" n copies ofn") is a partition in which a part of sizen, n ~ 1 can come n n different colours

, nn and parts sa.tisfy he orderenoted by subscripts: nl. n2.11 < 21 < 22 < ~ < 3.}< ~ < 41 < 42 < 43 < 4~ < 51 < 52.For example, here are six n-colourpartitions of 3, viz., ~,32, ~,21 + 11,22+11, i + 11 + 11Definition 4.(11]). An n-colour ordered partition is called an n-colour composi-tion. For example, here are eight n-colour compositionsof 3,viz., 3113~,331 ~11,

. C{ ~ V) d-ehO{.e. t..-e ?1u~Y1122,2111,1121,111111. l. 'A'J_I' c ('" - ~IJY CD'YY\'f1>\11VIl.:> OJLet C(v) denote the number of n-colour compositions of ~ nto m parts andC(m; q) and C(q) denote the enumerative generating functions for C(m, II} andC(v), respectively. The following basic formulas were proved in [1]:

?


3/9

qmG(mj q) = (1- q)2m' (1.1)C(q)= q1 - 39 + q2 '

lI+m-12m -IC(m, v) =

and

C(v) = F3vwhere F~v is the (2v)th Fibonacci number.2. An analogue of the Euler's partition identityWe shall prove the following result:Tbeorem 2.1.Let c(O, II) denote the nwnber of compositions of II into oddparts and cm 0, v) denote the number of compositions of v into exactly m oddparts. Let DE" denote he numberof "odd-even" artitions with largestpart vand DE: denote he numberof "odd-even" artitions with largestpart v intoexoctly m parts. Then

cm(o,v) DE': (2.1)and

c(O, II) = OEvo (2.2)Example. For 11= 6, m = 2, we see hat ~(O, 6) = 3, since here are 3 compo-sitions of 6 into 2 odd parts, viz., 5+1, 1+5, 3+3. Also, DE; = 3, since hereare 3 "odd-even"partitions into 2 parts with largest part 6, viz., 6+5. 6+3,6+1. FUrther,we see hat c(O,6) = 8, since here are 8 compositions f 6 intoodd parts viz., 5+1, 1+5, 3+3, 3+1+1+1, 1+3+1+1, 1+1+3+1, 1+1+1+3,

3


4/9

1+1+1+1+1+1. Also, OEe = 8, since he relevant"odd-even"partitions are6+5+4+3+2+1,6+5+4+3,6+5+4+1,6+5+2+1,6+3+2+1,6+5,6+3,6+1

We see hat (2.2) is very similar n structure o Euler's dentity.emark!We call (2.2) an analogue of Euler's identity for compositions.Remark 2. Obviously 2.2) is an immediateconsequencef (2.1). We shallprovide two proor;sfor 2.1).First proof of (2.1) (analytical). Let OEk(v) denote the Dumber of "odd-even" partitions of 1/ into m parts with largestpart k. Then by using MacMa-hon's partition analysiswe have

00~ OEr(lI)z*qVv=O}: z.2nl +mq(2nl +m)+t2~+m-I)+...+t2n...+I)

nl~~~...~n-?:O\nl-n~ \n~-n8 \n l-n...0"1 "2 ..0"m-I

=n~

where the variables AI, A~, .., Am-I handle the inequalities satisfied by nj whileThe linear operator n~ when applied o thehe nj themselves become free.

>'m-l annihilatesermswith any negative xponentsaurent series n -'1, -'2,

nm2 -!LAm-lm(m + 1)2 1m=z q

.1


5/9

Now on applying o eachof ~1,~2, ..,~m-l in Eq.(2.4) he following esult [4,Lemma.11.2.1, p.556]

we obta.inm(m + 1)2mq00L OE,:(v)z.q"1",=0 (1 - z~q~)(l - z~~)...(l - ~2q2rn)

By setting q =1 in (2.6), we obtain00L DE';' ;r!' =.=0

zm(1 - z~)m

On the other hand we see hat

00:):= c"'(0, v)z" = (z + z' + Z6+..=0 (2.8)zm.)m= (1 - z~)m

A comparision of (2.7) and (2.8) leads to (2.1)Remark.3. Using (2.7) and (2.8) one can easilyshow hat

00 00 qL OEnqn = L c(O,n)qn = 1- q - q2n=O n=O5


6/9

Since he extreme ight sideof (2.9) alsogenerates ibonaccinumbers, con-clude hat

DEn = c(D,n) = Fn (2.10)

YA Second proof of (2. 1) (combinatorial)1

00

j ~ xGraph A

Let Graph A be the graph of an "odd-even" partitiOIl1f into m parts with largestpart n. (In the graph A, n=6, m=4 and K=6+5+2+1, alsonote hat the x-axisis drawnoneunit of length below he last row and he y-axisoneunit of lengthto the left of the first column).We draw vertica.11inesrom the corner point of each part and measure he dis-tance of each ine from its preceding ne akillg y-axis also nto consideration.We see hat these distances give rise to a. composition of n into exactly m oddparts. Since the correspondence s one-to-one (2.1) is proved.The following is easily verified:

6


7/9

3+11+:1+:1+]-+-+-+-+

3. New combinatorial properties of n-colour compositionsWe shall prove the following results:Theorem 3.1. The numberof n-colourcompootions f 1/equals he numberof "odd-even" partitions with laz-gest az-t211.Theorem 3.2. The numberof n-colourcompootions f v equals he numberof compositions f 2v into odd parts.Theorem 3.3. Let~" denote the number of "odd~ven" partitions with largestpart odd and ~ 2v - 1. Then 6" equals he numberof n-colourcompositionsof II.

We haveearlier seen hat there are 85+4+3, 5+4+1, 5+2+1, 3, 3+2+1, 1n-colour compositions of 3.Theorem 3.4. The numberof n-colourcompositions f v equals he number

62 }4, 6442, 634 32,624222,643}2,66Theorem 3.5. The number of n-colour compositions of II equals he number ofpartitions into an evennumberof odd parts with largestpazot v - 1 such hatthe parts are alternately=: 3 and 1 (mod 4)

7

L+1J+JL+:l+l

L+lL+l~+1l +3.6+5+4+36+5+4+16+5+2+16+3+2+1

tation ( al a2 ... ar)aj alternate in parity.al ~ ... arExample. For 11= 3, there are 8 relevant s:elf-conjugate artitions viz.,6224,642212,


8/9

Example. For v = 3, here re relevantpartitions, viz.,11+9,11+5,11+1,11+9+7+5,11+9+7+1,11+9+3+1,11+5+3+1,11+9+7+5+3+1.Proofs of Theorems 3.1-3.5.Theorems 3.1-3.2 follow from (1.4) and (2.10).Theorem 3.3 follows from (1.4)(2.10)and he identity

(3.1)F2v-l = F2v.l + F3 +To proveTheorem3.4 we establisha bijection between he "odd-even"pacti-tions with largestpart 2v and he self-conjugateartitions with largestpart 2v

~ ) ; alternate in parity andthen use Theorem 3.1. We do it as follows:

,. > ar) be a.n "odd-even" partitionet 'If = a1 + a2 + ... + ar(211= a1 > a2 >with largest part 2v. We consider a graph which consists of r successive ends

.,ar-bend (here by a k-bend we Ir..eana right-bendvizo, at-bend, ~-bend,containingk dots in the first row as well as n the first column. For example,0 0 0a 3-bendmeans 0 ). We see mmediately hat this graph represents0self-conjugateartition with largestpart equal o 2v sum that in the Frobenius

Example. For II = 3, et us consider 7f=& + 5 + 2 + 1. Then

v v0 0which s a graphof the selfconjugate artition 624222 ith largestpart 6 such8

6+5+2+1 -+ 0000000000000 0 0 00 0 0 0


9/9

h - ., .-.' ..- (5 4 1 0 \ ,- .t at ll tne rrODenIUS nOtatiOn \ 5 4 1 0)' ~ alternate ll parIty.Theorem 3.5 follows from Theorem 3.4 once we observe hat if the right-bendsin the graphof a self-conjugateartition of Theorem3.4 are straightenedhenwe get a. paz-tition f Theorem3.5. For example, he self-conjugate artition624222 f the previousexamplecorrespondso the partition 11+9+3+1 whichis a.partition of the type described n Theorem 3.5.References1. A.K. Agarwal, n-colour compositions, ndian J.pure appl.Math, 31(11) (2000),1421-1427,2. A.K. Agaz-waland G.E. Andrews, Wgers-Ra.manujandentities, or paz-ti-tions with "N copies f N" , J. Combin. Theory Ser. A 45(1) (1987),40-49.3. G.E. Andrews,Rarnanujan's Lost" notebook V: Stacksand alternatingparity in partitions, Adv. Math. 53(1984),55-74.

G.E. Andrews,R. Askey and R. Roy, SpecialFUnctions,Encyclopedia f,Mathematics and its Applications, Vol. 71, Cambridge University press, 2000

G. Frobenius,Uber die Charaktereder symmetrischenGruppe, Sitzber..Preuss. Akad. Berlin, 1900, 51t>-534.6. G.H, Hardy and E.M. Wright, An Introduction to the Theory of Numbers,Oxford University Press, 1983.

~

agarwal a k - an analogue of euler's identity and new rial properties of n-colour composition - pp

Documents