Combinatorics of Necklaces and Hermite Reciprocity

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1 Joural of Algebraic Combiatorics 10 (1999), c 1999 Kluwer Acaemic Publishers. Maufacture i The Netherlas. Combiatorics of Necklaces a Hermite Reciprocity A. ELASHVILI alela@rmi.acet.ge M. JIBLADZE jib@rmi.acet.ge D. PATARAIA ito@rmi.acet.ge A. Razmaze Mathematical Istitute of the Georgia Acaemy of Scieces, M. Alexize St. 1, Tbilisi , Republic of Georgia Receive December 5, 1996; Revise March 26, 1998 Abstract. Combiatorial proof of a explicit formula for imesios of spaces of semi-ivariats of regular represetatios of fiite cyclic groups is obtaie. Usig bicolore ecklaces, a certai reciprocity law followig from this formula is also erive combiatorially. Keywors: partitio, ecklace, semi-ivariat, reciprocity Itrouctio The classical Hermite Reciprocity Law asserts the isomorphism S m S (k 2 ) = S S m (k 2 ) of symmetric powers of represetatios of the Lie group SL 2 (k) actig staarly o k 2, for a characteristic zero fiel k (see [6], Remark 12 by Popov i Appeix 3 of the Russia traslatio). I particular, the space of egree m polyomial ivariats of the irreucible ( + 1)-imesioal represetatio is equiimesioal with the space of egree ivariats of the irreucible (m + 1)-imesioal represetatio. Recetly i [3] there was obtaie a explicit formula for the imesio a 0 (, m) of the space of egree m homogeeous polyomial ivariats of the regular represetatio of the th orer cyclic group. This formula implies that a 0 (, m) = a 0 (m, ). I the preset paper, we give a combiatorial explaatio of a certai geeralizatio of this fact (see below), which we also call Hermite reciprocity. Relatioship with combiatorics stems from the observatio that, as show i [3], the umber a 0 (, m) coicies with the umber of solutios of the system 1 j=0 jλ j 0 (mo ); 1 λ i = m. (1) i=0 The first author is partially supporte by the INTAS Grat # The seco a thir authors are supporte by the ISF Grat MXH200 a by the INTAS Grat #

2 174 ELASHVILI, JIBLADZE AND PATARAIA Clearly this is the same as the total umber of partitios of multiples of ito o more tha m parts ot exceeig 1. Applyig combiatorial argumets oe ca obtai a formula for the umber a k (, m) of solutios of a eve more geeral system 1 j=0 jλ j k (mo ); 1 λ i = m (2) i=0 where k is ay oegative iteger. As a ki of illustratio let us reprouce the first few values of a k (, m) (compute usig the MAPLE package): (1) a 0 (, m), for 1, m 10: (2) a 1 (, m), for 1, m 10: Derivatio of a k (, m), give below, is aalogous to a proof for the a 0 (, m) commuicate to us by G. Arews. The expressio obtaie has all the avatages of a explicit formula, i particular it immeiately implies the equality a k (, m) = a k (m, ), which too

3 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 175 may be calle Hermite reciprocity. But the proof oes ot explai i ay way the reaso of this reciprocity. I the seco part of the paper we give oe of the possible explaatios for the equality. Namely: to ay solutio of (1) we assig a ecklace, i.e., a circular arragemet, cosistig of + m beas, of them black a m white, together with a chose orietatio a a basepoit somewhere betwee two ajacet beas. Thereafter, the equality a k (, m) = a k (m, ) turs out to follow from the existece of a ivolutio o the set of such ecklaces, actig by choosig opposite orietatio a swappig black a white. We must ote that i a private coversatio with the first author, N. Alo commuicate a proof ivolvig ecklaces, of a 0 (, m) = a 0 (m, ), whe a m are coprime. I the preset paper this iea has bee extee to the more geeral settig. The authors woul like to express their gratitue to Alo, Arews a Staley for valuable iformatio a iterest to the paper. They are grateful to the referee for careful reaig of the paper a fiig of several misprits i importat formulae. Everywhere i the sequel, for ay itegers, m, k,... their greatest commo ivisor will be eote by (, m, k,...); for > 0, we eote by (k) the resiue of k moulo, i.e., the umber etermie by 0 (k) <, (k) k(mo ). 1. Explicit formula Let us start with a purely formal expressio for a k (, m). Therefore recall that p(n, M, s), for ay itegers N, M, s, eotes the umber of partitios of s ito o more tha M parts, each ot exceeig N. The geeratig fuctio for these umbers, G(N, M; t) = s p(n, M, s)t s is the Gauss polyomial (see e.g., [1], 3.2). The, oe obviously has a k (, m) = j p( 1, m, j +k). (3) We shall also ee the efiitio of Ramauja sums (see e.g., [4], 17.6; for applicatios i umber theory see [5]). For ay a k, the Ramauja sum c (k) is the sum of kth powers of all primitive th roots of 1. I particular, c (0) = ϕ() (the Euler fuctio), c (1) = µ() (the Möbius fuctio). It is kow (a easily see usig Möbius iversio) that c (k) = (,k) ( ) µ. Also ote that this last equality obviously implies c (k) = c ((, k)), i particular, c ( k) = c (k). We the have the followig:

4 176 ELASHVILI, JIBLADZE AND PATARAIA Theorem 1 For ay itegers k,, m, ( ) a k (, m) = 1 / + m/ c (k) ; (4) + m / i particular a k (, m) = a k (m, ). (,m) Proof: By (3), a k (, m) equals the sum of coefficiets of G( 1, m; t) at those powers of t which are cogruet to k moulo. Now i geeral, give ay polyomial f (t) = f ν t ν, oe has f ν = 1 ζ k f (ζ ), ν k (mo ) ζ =1 the sum o the right ruig over all th roots of 1. This fact easily follows from the equality { if ν ζ ν = 0 otherwise. ζ =1 So i our case a k (, m) = 1 ζ k G( 1, m; ζ). ζ =1 Values of Gauss polyomials at roots of 1 are kow; see e.g., [7], Chapter 3, Exercise 45(b). I particular, for ay ζ = 1 which is a primitive th root of 1, for some, oe has ( ) m/ +/ 1 if m G( 1, m; ζ) = m/ 0 otherwise. Hece ( ) a k (, m) = 1 m/ + / 1 ζ k, m/ or(ζ )= m where or(ζ ) meas orer of the elemet ζ i the group of roots of 1. Now ζ has orer iff ζ is a primitive root of orer. Hece ( ) a k (, m) = 1 m/ + / 1 c ( k) m/ (,m) = 1 ( m+ 1 )! c (k)( 1)! m! (,m)

5 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 177 = 1 = 1 (,m) (,m) = 1 m + c (k) m+! m+! m! ( ) m/ + / c (k) m + / (,m) ( ) m/ + / c (k), / a the theorem follows. Remark Note that the formula obtaie implies, ue to the metioe properties of Ramauja sums, that there are may equalities betwee a k (, m) for fixe, m a various k. Namely, oe has a k (, m) = a (k,,m) (, m). For the sequel, let us fix two positive itegers, m a eote (, m) by. 2. Combiatorial proof of reciprocity I this sectio, we are goig to give aother, purely combiatorial proof of the equality a k (, m) = a k (m, ). Cosier the set { } 1,m = (λ 0,...,λ 1 ) N iλ i 0a λ i = m. We efie a actio of the cyclic group C = Z/Z of orer o,m by settig, for r Z/Z a λ = (λ 0,...,λ 1 ),m, rλ=(λ r,λ r+1,...,λ 1,λ 0,...,λ r 1 ). For ay λ,m, eote by t(λ) the miimal positive iteger with (t(λ)1)λ = λ, i.e., the umber of elemets i the orbit of λ uer the actio of C (1 is the elemet of Z/Z ={0,1,..., 1}). Deote /t(λ), i.e., orer of the stabilizer of λ uer this actio, by s(λ). Let,m be the quotiet,m /C a eote by π,m the quotiet map π,m :,m,m. Take ay α,m a λ π 1,m i=0 1 (α). Note that #(π,m(α)) = t(λ). Sice s(λ) oes ot epe o the choice of the iverse image of α, we may eote by s(α) the umber s(λ) for ay λ π 1,m (α).

6 178 ELASHVILI, JIBLADZE AND PATARAIA Let K,m be the mappig from,m to C etermie by K,m (λ 0,...,λ 1 )=(0 λ 0 +1 λ 1 + +( 1) λ 1 ). The it is clear that a k (, m) equals #(K 1,m (k)), for ay k C. Propositio 1 For ay r C,λ = (λ 0,...,λ 1 ),m oe has K,m (rλ) = K,m (λ) + r (m). Proof: The case r = 0 is obvious. For r = 1, oe has K,m (1 (λ 0,...,λ 1 )) = K,m (λ 1,λ 0,...,λ 2 ) = (0 λ 1 +1 λ 0 + +( 1)λ 2 ) For 1 < r <, oe has = (0 λ λ 1 + +( 2)λ 2 + ( 1)λ 1 + (λ 0 + +λ 2 ( 1)λ 1 )) = K,m (λ) + (λ 0 + +λ 2 +λ 1 λ 1 ) = K,m (λ) + 1 (m). K,m (rλ) = K,m (1...(1 λ)...)) = K }{{},m (λ) + (m) + +(m) }{{} r times r times = K,m (λ) + r (m) Deote by p the atural projectio from C to C ; for r Z/Z ={0,..., 1}, p (r)=(r). Sice ivies, this is clearly a group homomorphism. Defie ow the mappig K,m :,m C by assigig to α,m the elemet p (K,m(λ)) C, where λ is ay elemet of π,m 1 (α). Sice ivies m, by Propositio 1 the elemet p (K,m(λ)) oes ot epe o the choice of λ. So oe obtais a commutative iagram,m π,m K,m C p (5),m K,m C.

7 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 179 Propositio 2 Let α,m a K,m (α) = r C. The for ay r (p ) 1 (r) there are exactly /s(α) elemets i π,m 1 (α),m which are mappe to r by K,m. Proof: Cosier ay λ π,m 1 (α). By commutativity of (5), clearly p K,m(λ) = r. By Propositio 1, oe has K,m ((l 1)λ) = K,m (λ) + l (m) for ay l. Those l for which K,m ((l 1)λ) = K,m (λ), are precisely those for which l (m) = 0i Z/Z. Sice (, m) =, those l must be ivisible by /. So the miimal ozero l with K,m ((l 1)λ) = K,m (λ) is /. Hece for ay istict l 1, l 2 from the iterval [0; /[ oe has K,m ((l 1 1)λ) K,m ((l 2 1)λ). This implies, firstly, that t(λ) is ivisible by /, i.e., t(λ)/ = /s(λ) is iteger, a secoly, sice the iverse image of r uer p has / elemets, that whe l rus over the iterval [0; /], the K,m ((l 1)λ) will become each elemet of the iverse image of r uer p exactly oce. This meas that, for each r (p ) 1 (r), the umber of those 0 l < with K,m ((l 1)λ) = r equals. Hece the umber of those λ from π 1,m (α) with K,m(λ) = r equals /s(α). Cosier ow the set W+m, whose elemets are circular arragemets ( ecklaces ) of + m beas, of them black a m white. There has to be fixe orietatio of the circle, as well as a basepoit locate somewhere betwee two ajacet beas. Let ϒ+m be the subset of W +m cosistig of those arragemets for which the first bea alog the orietatio after the basepoit is black. Let us efie a actio of C o ϒ+m as follows: for β ϒ +m a r C, rβ will be the same arragemet as β but with the basepoit shifte couterorietatiowise exactly by the amout eee for the umber of passe black beas to become r. Deote ϒ+m /C by ϒ +m ;so ϒ +m is the set of arragemets as above, without ay basepoit, a cosiere up to rotatio. The atural projectio from ϒ+m to ϒ +m will be eote by π +m. For β ϒ+m, we eote by t(β) the miimal positive iteger with (t(β) 1)β = β. Clearly t(β) = t(rβ) for ay β ϒ+m a r C. The umber /t(β) will be eote by s(β). For ay γ ϒ +m, the umber s(β), for β (π +m ) 1 (γ ) oes ot epe o the choice of β; we will eote this umber by s(γ ). Let us costruct a map g+m : ϒ +m C. Take β ϒ+m a suppose that umbers of black beas i β, coute orietatiowise from the basepoit, are 1, r 2,...,r (by efiitio the first bea is black). The oe efies g+m (β) to be the elemet (1 + r 2 + +r ) (1+2+ +) of C. Now costruct the map g +m : ϒ +m C. Give ay γ ϒ +m, choose a basepoit o it somewhere betwee two ajacet beas. Suppose the umbers of the black beas coute orietatiowise w. r. t. this basepoit are r 1,...,r. Cosier the umber r 1 + +r. If oe woul choose a ifferet basepoit, each of the r i woul chage to r i i such a way

8 180 ELASHVILI, JIBLADZE AND PATARAIA that (r i r i ) +m woul be the same for all i; eote this resiue by. The (r 1 + +r ) +m =(r 1 + +r ) +m + a p +m ((r 1 + +r ) +m ) = p +m ((r 1 + +r ) +m)+ p +m ( ) = p +m ((r 1 + +r ) +m). So we may efie g +m = (r 1 + +r ) (1+ +). It is clear that the iagram ϒ +m g +m C π +m p ϒ +m g +m C commutes. Let us ow costruct a map w :,m ϒ +m. For λ = (λ 0,...,λ 1 ),m choose a orietatio a a basepoit o a circle; start movig from the basepoit orietatiowise a put the first black bea. The put ext λ 1 white beas a the ext black oe; agai put λ 2 white beas a the ext black oe a so o. O the ( 1)-th step, whe we will put λ 0 white beas there will be + m beas arrage the ext oe will be the black bea we starte with. So oe obtais a elemet of ϒ +m which we efie to be w(λ 0,...,λ 1 ). It is easy to see that w is a bijectio compatible with the actio of C. Moreover oe has the followig propositio. Propositio 3 The iagram,m w ϒ +m K,m g +m Z/Z commutes. Proof: Take (λ 0,...,λ 1 ),m. The i w(λ 0,...,λ 1 ), umbers of black beas coute from the basepoit orietatiowise will be 1, λ 1 +2,λ 2 +λ 1 +3,...,λ 1 + +λ 1 +.

9 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 181 Hece g +m (w(λ 0,...,λ 1 )) = = (1 + (λ 1 + 2) + (λ 2 + λ 1 + 3) + +(λ 1 + +λ 1 +)) (1 + +) = (( 1)λ 1 + ( 2)λ 2 + +λ 1 ) = (0 λ 0 + +( 1)λ 1 ) = K,m (λ 0,...,λ 1 ). Sice the bijectio w commutes with the actio of C, it iuces a bijectio w :,m ϒ +m, a moreover by Propositio 3 there is a commutative iagram,m π,m,m K,m w ϒ + m g +m π + m w ϒ + m g +m K,m (*) C p C Propositio 4 For ay r C, # ( K 1,m (r)) = γ ( g +m) 1 (p (r)) s(γ ). Proof: This is obvious from Propositio 2 a commutativity of (*). Let us ow costruct the map x : ϒ +m ϒ m +m as follows: for γ ϒ +m, efie the elemet x(γ ) ϒ +m m by reversig the orietatio a chagig black beas by white oes a vice versa.

10 182 ELASHVILI, JIBLADZE AND PATARAIA Propositio 5 The iagram ϒ +m x ϒ m +m commutes. g +m g m +m Z/Z Proof: Take ay γ ϒ +m a choose i it a basepoit betwee some ajacet beas. Choose the same basepoit i x(γ ). Suppose that i γ the umbers of black beas are b 1,...,b a those of white oes are c 1,...,c m. Sice each bea is either white or black, i.e., b 1 + +b +c 1 + +c m =1+ +(+m), c 1 + +c m =1+ +(+m) (b 1 + +b ). (6) Numbers of black beas i x(γ ) w.r.t. the ew orietatio will be + m + 1 c 1,...,+ m+1 c m. Hece g m +m (x(γ )) = ( + m + 1 c m+1 c m ) (1+ +m) = (m(+m+1) c 1 c m ) (1+ +m) = (c 1 + +c m ) (1+ +m) ( m) = (b 1 + +b ) (1+ +(+m)) (1 + +m) (by (6)) ( ) (+m)( + m + 1) m(m + 1) = (b 1 + +b ) ( = (b 1 + +b ) m(+m+1)+ (+1) ) 2 ( ) (+1) = (b 1 + +b ) ( m) 2 = (b 1 + +b ) (1+ +) = g+m (γ ). We have reache the goal of this sectio. Proof of a k (, m) = a k (m, ): Take ay C a m C m with ( ) = (m ) = k. By Propositio 4, # ( K 1,m ( ) ) = γ ϒ +m g +m (γ )= s(γ ).

11 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 183 Hece usig the isomorphism x : ϒ +m ϒ +m m oe obtais # ( K 1,m ( ) ) = It follows that = γ ϒ +m g +m (γ )= γ ϒ +m g +m m (x(γ ))= s(γ ) = γ ϒ +m g +m (γ )= s(x(γ )) = s(x(γ )) δ ϒ m +m g m +m (δ)= s(δ) = #( K 1 m, (m ) ). a k (, m) = # ( K 1,m ( ) ) = # ( K 1 m, (m ) ) = a k (m, ). 3. Aother proof of the formula Our ext task will be to obtai aother erivatio of the formula 4, a k (, m) = 1 + m +m! c (k)! m!, of a more combiatorial ature. Cosier the actio of C +m o the set W+m, uer which r C +m acts by shiftig the basepoit by r beas couterorietatiowise. Let W+m (k), for k C, be the subset of W+m cosistig of those elemets with ( ) r i i = k, i=1 i=1 where r 1,...,r are umbers of black beas coute orietatiowise from the basepoit. Propositio 6 W+m. The subset W +m (k) W +m is ivariat uer the actio of C +m o Proof: Take r C +m, a let γ be a elemet of W+m (k), umbers of black beas of γ beig r 1,...,r. The the umbers of black beas i rγ W+m will be r ε 1( +m)+r 1, r ε 2 ( + m) + r 2,...,r ε (+m)+r, where each ε i is 0 or 1 epeig o whether r i + r < + m or ot (i other wors, umbers of those beas ot passe by the basepoit will grow by r, while of those passe by r ( + m)). Hece the sum of umbers of black beas will become r ε i ( + m) + r i = r ( + m) i=1 ε i + i=1 r i. i=1

12 184 ELASHVILI, JIBLADZE AND PATARAIA So sice both a +m are ivisible by, the sum of umbers of black beas will remai the same moulo. Deote W+m (k)/c +m by W +m (k). Clearly W +m (k) ={γ ϒ +m g +m (γ ) = k}. The iverse image i W+m (k) of each γ W +m +m (k) has elemets, sice the patter s(γ ) of the arragemet γ is perioic with perio +m +m a there are possibilities to choose s(γ ) s(γ ) the basepoit. Hece # ( W +m (k)) = + m s(γ ). γ ϒ +m g +m m (γ )=k Comparig this with the formula from Propositio 4, # ( K 1,m ( ) ) = γ ϒ +m g +m (γ )=k s(γ ), oe coclues # ( K,m 1 ( ) ) = + m #( W +m (k)). (7) Let us calculate #(W+m (k)). Propositio 7 # ( W+m (k)) = 1 c (k) ( +m ). Proof: Cosier the polyomial P(x, t) = x (+1) 2 k (t + x)(t + x 2 ) (t +x +m ). After expaig P(x, t) a collectig the terms, the coefficiet at the moomial t m x l will be the umber of represetatios of t m x l = x (+1) 2 k t m x r 1 x r2 x r, where 0 < r 1 < < r +m with ( + 1) 2 k + j r j = l. Rewrite the last equality as r j j ( + 1) 2 k = l ( + 1);

13 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 185 sice ivies ( + 1), it is clear that the sum of these coefficiets at t m x l over all l with l will equal the umber of those sequeces 0 < r 1 < <r +m with ( r j j ) ( + 1) = k. 2 Assigig to such a sequece a arragemet from W+m with r 1,...,r as umbers of black beas, shows that the sum of these coefficiets equals #(W+m (k)). Let ζ = e 2πi (or ay other primitive th root of 1). Sice for ay iteger l oe has { if l ζ lj = 0 otherwise, j=1 it follows that #(W +m (k)) equals the coefficiet at t m of the polyomial P(t) = 1 P(ζ j, t). j=1 Cosier ow the polyomials P(ζ j, t) separately. Sice, oe has ζ j (+1) 2 = { 1 if is o or both a j 1 if is eve a j is o, i.e., ζ j (+1) 2 = ( 1) ( 1)j. Hece are eve, P(ζ j, t) = ( 1) ( 1)j ζ jk (t +ζ j )(t + ζ 2 j ) (t +ζ (+m)j ). Deotig t by s, oe obtais P(ζ j, t) = ( 1) ( 1)j ζ jk ( 1) +m( (s ζ j ) (s ζ (+m)(, j) (,j) j)) = ( 1) ( 1)j ++m ζ jk( s )(+m)(, j) (,j) 1. Now collect together the terms of P(t) = 1 P(ζ j, t) whose j s have the same gc with. Oe obtais P(t) = ( 1)+m ( s 1 ) (+m) ( j,)= ( 1) ( 1)j ζ jk.

14 186 ELASHVILI, JIBLADZE AND PATARAIA The sig term i the last sum is ( 1) ( 1)j = = { 1 if is eve a both a j are o, 1 otherwise, { 1 if is eve a both a = ( j, ) are o, 1 otherwise, = ( 1) ( 1) /. Hece P(t) = ( 1)+m = ( 1)+m = ( 1)+m ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) So the coefficiet at t m i P(t) will be # ( W +m (k)) = ( 1)+m = ( 1) (s 1) (+m) ( j,)= ζ jk (s 1) (+m) c ( k) c ( k)(( 1) t 1) (+m). ( 1) ( 1) ( 1) c ( k) c ( k)( 1) m ( 1) ( ) +m ( +m ). Sice, the sig term i that last sum is ( 1) = ( 1). Hece # ( W+m (k)) = 1 c (k) ( +m ) (we have use the eviet equality c ( k) = c (k)). Remark A similar argumet for erivig a 0 (, m) has bee commuicate to us by Alo a Staley: for the role playe by our P(x, t), they use the fuctio (1 tx) 1 (1 tx 2 ) 1 (1 tx 1 ) 1.

15 COMBINATORICS OF NECKLACES AND HERMITE RECIPROCITY 187 We have reache the esire formula: iee, from (7) a a k (, m) = #(K,m 1 (k)) oe immeiately obtais a k (, m) = 1 + m c (k) ( +m ). 4. Some Fial Remarks Geeratig fuctios for a 0 (, m) metioe i [3] ca be easily geeralize: for ay iteger k oe has a,m=0 ζ k (s + t) a k (, m)x y m = 1,m=1 =1 c (k) log(1 x y ) a k (, m) s m t = ζ(s +t +1),m=1 ( +m 1)! s m t,! m! where ζ k (x) = k x (i particular ζ 0 (x) = ζ(x) is the Riema zeta fuctio). Let us ote also a relatioship of a k (, m) to free Lie algebras. Cosier a free Lie algebra Lie(x, y) o two geerators x, y over a characteristic zero fiel. The Th. 2(b) i [2], Ch. II, Sectio 3, o. 3 immeiately implies that for ay, m, a 1 (, m) = im(lie(x, y),m ), where (),m eotes the homogeeous compoet of biegree i x a m i y. We coul ot fi a explicit correspoece betwee geerators of the above Lie algebra a our ecklace iterpretatio of a 1 (, m). This coectio looks eve more iterestig i view of the fact that i a sese, all the a k may be reuce to a 1. Propositio 8 For ay, m, a k, a k (, m) = a 1 (/, m/ ). Proof: (,m,k) Oe calculates irectly: (,m,k) a 1 (/, m/ ) = (,m,k) = (,m) 1 / + m/ 1 + m (,k) (/,m/ ) µ(/ ) ( +m µ( ) ) ( ) +m

16 188 ELASHVILI, JIBLADZE AND PATARAIA = 1 + m (,m) c (k) ( +m ) = a k (, m). We o ot kow a combiatorial explaatio of this fact, either. Refereces 1. G.E. Arews, The theory of partitios, Ecyclopeia of Mathematics a Its Applicatios, G.-C. Rota, (E.), Aiso-Wesley, Reaig, MA, 1976, Vol N. Bourbaki, Groupes et algébres e Lie, Herma, Paris, A. Elashvili a M. Jiblaze, Hermite Reciprocity for regular represetatios of cyclic groups, Iag Mathem., N.S., 9(2), 1998, Higher Trasceetal Fuctios, Batema Mauscript Project, A. Erélyi (ir.), McGraw-Hill, New York, S. Ramauja, O certai trigoometrical sums a their applicatios i the theory of umbers, Tras. Camb. Phil. Soc., XXII(13) (1918), T. A. Spriger, Ivariat theory, Lecture Notes i Math., Vol. 585, Spriger-Verlag, Berli a New York, R. Staley, Eumerative Combiatorics, Vol. I, Wasworth & Brooks/Cole, Moterey, CA, 1986.

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions

6.451 Principles of Digital Communication II Wednesday, March 9, 2005 MIT, Spring 2005 Handout #12. Problem Set 5 Solutions 6.51 Priciples of Digital Commuicatio II Weesay, March 9, 2005 MIT, Sprig 2005 Haout #12 Problem Set 5 Solutios Problem 5.1 (Eucliea ivisio algorithm). (a) For the set F[x] of polyomials over ay fiel F,