Alo available a hp://amc-journal.eu ISSN 1855-3966 (prined edn.), ISSN 1855-3974 (elecronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 417 423 The muliube um problem for finie abelian group Amela Muraović-Ribić Univeriy of Sarajevo, Deparmen of Mahemaic, Zmaja od Bone 33-35, 71000 Sarajevo, Bonia and Herzegovina Qiang Wang School of Mahemaic and Saiic, Carleon Univeriy, 1125 Colonel By Drive, Oawa, Onario, K1S 5B6, Canada Received 28 Ocober 2013, acceped 29 Augu 2014, publihed online 11 June 2015 Abrac We ue a imilar echique a in [2] o derive a formula for he number of muliube of a finie abelian group G wih any given ize and any given mulipliciy uch ha he um i equal o a given elemen g G. Thi alo give he number of pariion of g ino a given number of par over a finie abelian group. Keyword: Compoiion, pariion, ube um, polynomial, finie field, characer, finie abelian group. Mah. Subj. Cla.: 11B30, 05A15, 20K01, 11T06 1 Inroducion Le G be a finie abelian group of ize n and D be a ube of G. The well nown ube um problem in combinaoric i o decide wheher here exi a ube S of D which um o a given elemen in G. Thi problem i an imporan problem in complexiy heory and crypography and i i NP-complee (ee for example [3]). For any g G and i a poiive ineger, we le he number of ube S of D of ize i which um up o g be denoed by N(D, i, g) = #{S D : #S = i, S = g}. Reearch i parially uppored by NSERC of Canada. E-mail addree: amela@pmf.una.ba (Amela Muraović-Ribić), wang@mah.carleon.ca (Qiang Wang) cb Thi wor i licened under hp://creaivecommon.org/licene/by/3.0/
418 Ar Mah. Conemp. 8 (2015) 417 423 When D ha more rucure, Li and Wan made ome imporan progre in couning hee ube um by a ieve echnique [3, 4]. Recenly Koer [2] give a horer proof of he formula obained by Li and Wan earlier, uing characer heory. N(G, i, g) = 1 n ( 1) i+ d gcd(e(g),) µ(/d)#g[d], where exp(g) i he exponen of G, e(g) = max{d : d exp(g), g dg}, µ i he More generally, we conider a muliube M of D. The number of ime an elemen belong o M i he mulipliciy of ha member. We define he mulipliciy of a muliube M i he large mulipliciy among all he member in M. We denoe M(D, i, j, g) = #{muliube M of D : mulipliciy(m) j, #M = i, M = g}. I i an inereing queion by i own o coun M(D, i, j, g), he number of muliube of D of cardinaliy i which um o g where every elemen i repeaed a mo j ime. If j = 1, hen M(D, i, j, g) = N(D, i, g). If j i, hi problem i alo equivalen o couning pariion of g wih a mo i par over D, which i M(D, i, i, g). In hi cae we ue a impler noaion M(D, i, g) becaue he econd i doe no give any rericion. Anoher moivaion o udy he enumeraion of muliube um i due o a recen udy of polynomial of precribed range over a finie field. Indeed, hrough he udy of enumeraion of muliube um over finie field [5], we were able o diprove a conjecure of polynomial of precribed range ove a finie field propoed in [1]. Le F q be a finie field of q elemen and F q be he cyclic muliplicaive group. When D i F q (he addiive group) or F q, couning he muliube um problem i he ame a couning pariion over finie field, which ha been udied earlier in [6]. In hi noe, we ue he imilar mehod a in [2] o obain M(D, i, j, g) when D = G. However, we wor in a power erie ring inead of a polynomial ring. Theorem 1. Le G be a finie abelian group of ize n and le g G, i, j Z wih i 0 and j 1. For any n, we define Then we have C(n, i, j, ) = M(G, i, j, g) = 1 n 0,0 n gcd(,j+1), + lcm(,j+1)=i ( 1) ( n/ + 1 C(n, i, j, ) )( n gcd(,j+1) ). µ(/d)#g[d]. where exp(g) i he exponen of G, e(g) = max{d : d exp(g), g dg}, µ i he A a corollary, we obain he main heorem in [2] when j = 1.
A. Muraović-Ribić, Q. Wang: The muliube um problem for... 419 Corollary 1. (Theorem 1.1 in [2]) Le G be a finie abelian group of ize n and le g G and i Z. Then we have N(G, i, g) = 1 ( 1) i+ µ(/d)#g[d]. n where exp(g) i he exponen of G, e(g) = max{d : d exp(g), g dg}, µ i he Moreover, when j i, he formula give he number of pariion of g wih a mo i par over a finie abelian group. To avoid confuion he mulie coniing of a 1,..., a n i denoed by {{a 1,..., a n }}, wih poibly repeaed elemen, and by {a 1,..., a n } he uual e. We define a pariion of he elemen g G wih exacly i par in D a a mulie {{a 1, a 2,..., a i }} uch ha all a are nonzero elemen in D and a 1 + a 2 +... + a i = g. Then he number of hee pariion i denoed by P D (i, g), i.e., {. P D (i, g) = {{a 1, a 2,..., a i }} D : a 1 + a 2 +... + a i = g, a 1,..., a i 0} I urn ou M(D, i, g) = i =0 P D(, g) i he number of pariion of g G wih a mo i par in D. Corollary 2. Le G be a finie abelian group of ize n and le g G. Then he number of pariion of g over G wih a mo i par i 1 ( ) n/ + 1 µ(/d)#g[d]. n where exp(g) i he exponen of G, e(g) = max{d : d exp(g), g dg}, µ i he Proof. The number i M(G, i, j, g) when j i 0. If j i, hen he linear Diophanine equaion + lcm(, j + 1) = i reduce o = i and = 0. The re of proof follow immediaely. In Secion 2, we prove our main heorem and derive Corollary 1 a a conequence. In Secion 3, we exend our udy o a ube of a finie abelian group and mae a few remar on how o obain he number of pariion over any ube of a finie abelian group. 2 Proof of Theorem 1 To mae hi paper elf-conained, we recall he following lemma (ee Lemma 2.1-2.4 in [2]). Le G be a finie abelian group of ize n. Le C be he field of complex number and Ĝ = Hom(G, C ) be he group of characer of G. Le χ Ĝ and χ be he conjugae characer which aifie χ(g) = χ(g) = χ( g) for all g G. We noe ha a characer χ can be naurally exended o a C-algebra morphim χ : C[G] C on he group ring C[G]. Lemma 1. Le α = g G α gg C[G]. Then we have α g = 1 n χ Ĝ χ(g)χ(α).
420 Ar Mah. Conemp. 8 (2015) 417 423 Lemma 2. Le m be a poiive ineger and g G. Then χ(g) = δ g mg #G[m], χ Ĝ,χm =1 where δ g mg i 1 if g mg and i i zero oherwie. Lemma 3. Le χ Ĝ be a characer and m be i order. Then we have (1 χ(σ)y ) = (1 Y m ) n/m. σ G Lemma 4. Le g G. The number e(g) i equal o lcm{d : d exp(g), g dg}. For d exp(g) we have g dg if and only if d e(g). Le u preen he proof of Theorem 1. We ue he muliplicaive noaion for he group. Proof. Fix j 1. Woring in he power erie ring C[G][[X]] over he group ring, he generaing funcion of g G M(G, i, j, g)g i g GM(G, i, j, g)gx i = σ G(1+σX+ +σ j X j ) = σ G 1 σ j+1 X j+1 1 σx C[G][[X]]. Uing Lemma 1, we wrie M(G, i, j, g)x i = 1 χ(g) n σ G χ Ĝ Separaing he fir um on he righ hand ide, we obain 1 χ j+1 (σ)x j+1 1 χ(σ)x. M(G, i, j, g)x i = 1 n χ(g) exp(g) χ Ĝ,ord(χ)= σ G 1 χ j+1 (σ)x j+1 1 χ(σ)x. For each fixed χ of he order, we now ha χ j+1 ha he order gcd(,j+1). Therefore by Lemma 3, we implify he above a follow: M(G, i, j, g)x i = 1 n Noe ha χ(g) exp(g) χ Ĝ,ord(χ)= χ Ĝ,χ =1 χ(g) = d ( 1 X lcm(,j+1) ) n gcd(,j+1) (1 X ) n/. (2.1) χ Ĝ,ord(χ)=d χ(g). By Lemma 2 and he Möbiu inverion formula, we obain χ(g) = µ(/d) χ(g) = µ(/d)δ g dg #G[d]. χ Ĝ,ord(χ)= d d χ Ĝ, χd =1
A. Muraović-Ribić, Q. Wang: The muliube um problem for... 421 Becaue d exp(g), by Lemma 4, g dg if and only if d e(g). Hence χ(g) = µ(/d)δ g dg #G[d] = µ(/d)#g[d]. χ Ĝ,ord(χ)= d Plugging hi ino Equaion (2.1), we ge M(G, i, j, g)x i = 1 n exp(g) µ(/d)#g[d] ( 1 X lcm(,j+1) ) n gcd(,j+1) (1 X ) n/. By applying he binomial heorem o he righ hand ide and comparing coefficien of X i in boh ide, we ingle ou M(G, i, j, g) and obain M(G, i, j, g) = 1 n exp(g) µ(/d)#g[d]c(n, i, j, ). Afer bringing C(n, i, j, ) ou of he inner um we complee he proof. Finally we remar ha we can derive Corollary 1 uing N(G, i, g) = M(G, i, 1, g). When j = 1, le u conider + lcm(, j + 1) = + lcm(, 2) = i. If i even, we obain + = i and hu + =. Noe ha we have he following power erie expanion and 1 (1 x) = n/ =0 =0 ( n/ + 1 ) x, 2n/ ( ) 2n/ (1 x) 2n/ = ( 1) x, n/ (1 x) n/ = ( 1) j x j. j j=0 Now we compare he coefficien of he erm x in boh ide of 1 (1 x ) n/ (1 x ) 2n/ = (1 x ) n/, afer expanding hee power erie. Hence we obain C(n, i, 1, ) = += 0,0 2n/ ( 1) ( n/ + 1 )( ) 2n/ = ( 1). Moreover, C(n, i, 1, ) = ( 1) i+( n/ ) becaue i i even. Similarly, if i odd, we obain + 2 = i and hu + 2 =. Moreover, i + i even. Uing (1 x 2 ) n/ 1 (1 x ) n/ = (1 + x ) n/,
422 Ar Mah. Conemp. 8 (2015) 417 423 we obain C(n, i, 1, ) = 3 A few remar +2= 0,0 n/ ( 1) ( n/ + 1 ) = ( 1) i+. In hi ecion we udy M(D, i, j, g) where j i and D i a ube of G. We recall ha in hi cae we ue he noaion M(D, i, g) becaue j doe no really pu any rericion. Fir of all, we noe ha M(G \ {0}, i, g)gx i = g G By Corollary 2, we obain M(G \ {0}, i, g) = 1 n gcd(exp(g),i 1) σ G,σ 0 ( ) n/ + 1 1 1 σx = (1 X) ( ) n/ + (i 1)/ 1 (i 1)/ M(G, i, g)gx i. g G µ(/d)#g[d] µ(/d)#g[d]. We noe M(G \ {0}, i, g) = P G (i, g). Therefore we obain an explici formula for he number of pariion of g ino i par over G. More generally, le D = G \ S, where S = {u 1, u 2,..., u S } =. Denoe by M S (G, i, g) he number of muliube of G of ize i ha conain a lea one elemen from S. Then he number of muliube of D = G \ S wih i par which um up o g i equal o M(G \ S, i, g) = M(G, i, g) M S (G, i, g). Noe ha M(G, 0, 0) = 1 and M(G, 0, ) = 0 for any G \ {0}. The principle of incluion-excluion immediaely implie ha M S (G, i, g) i given in he following formula. We noe ha he formula i quie ueful when he ize of S i mall in order o compue M(G \ S, i, g). Propoiion 1. For all i = 1, 2,... and g G we have +( 1) 1 +( 1) i 2 ( 1) i 1 M S (G, i, g) = u S M(G, i 1, g u)... {u 1,u 2,...,u } S {u 1,u 2,...,u i 1} S {u 1,u 2,...,u i} S M(G, i, g (u 1 + u 2 +... + u )) +... M(G, 1, g (u 1 + u 2 +... + u i 1 ))+ M(G, 1, g (u 1 + u 2 +... + u i )).
A. Muraović-Ribić, Q. Wang: The muliube um problem for... 423 Proof. Fix an elemen g G. Denoe by A u he family of all he muliube of G wih i par which um up o g and each muliube alo conain he elemen u. The principle of he incluion-excluion implie ha u S A u = A u A u1 A u2 +... (3.1) u S {u 1,u 2} S I i obviou o ee A u1 A u2 = M(G, i 2, g (u 1 + u 2 )) ec. by definiion and he reul follow direcly. Acnowledgemen We han anonymou referee for heir helpful uggeion. Reference [1] A. Gác, T. Héger, Z. L. Nagy, D. Pálvölgyi, Permuaion, hyperplane and polynomial over finie field, Finie Field Appl. 16 (2010), 301-314. [2] M. Koer, The ube problem for finie abelian group, J. Combin. Theory Ser. A 120 (2013), 527-530. [3] J. Li and D. Wan, On he ube um problem over finie field, Finie Field Appl. 14 (2008), 911-929. [4] J. Li and D. Wan, Couning ube um of finie abelian group, J. Combin. Theory Ser. A 119 (2012), no. 1, 170-182. [5] A. Muraović-Ribić and Q. Wang, On a conjecure of polynomial wih precribed range, Finie Field Appl. 18 (2012), no. 4, 728-737. [6] A. Muraović-Ribić and Q. Wang, Pariion and compoiion over finie field, Elecron. J. Combin. 20 (2013), no. 1, P34, 1-14.