On an identity for the cycle indices of rooted tree automorphism groups

Transcription

1 On an dentty for the cycle ndces of rooted tree autoorphs groups Stephan G Wagner Insttut für Analyss und Coputatonal Nuber Theory Technsche Unverstät Graz Steyrergasse 30, 800 Graz, Austra wagner@fnanzathtugrazat Subtted: Jul 25, 2006; Accepted: Sep 5, 2006; Publshed: Sep 22, 2006 Matheatcs Subject Classfcatons: 05A5,05A9,05C30 Abstract Ths note deals wth a forula due to G Labelle for the sued cycle ndces of all rooted trees, whch resebles the well-nown forula for the cycle ndex of the syetrc group n soe way An eleentary proof s provded as well as soe edate corollares and applcatons, n partcular a new applcaton to the enueraton of -decoposable trees A tree s called -decoposable n ths context f t has a spannng forest whose coponents are all of sze Introducton Pólya s enueraton ethod s wdely used for graph enueraton probles we refer to [6] and the references theren for nstance For the applcaton of ths ethod, nforaton on the cycle ndces of certan groups s needed ostly, these are coparatvely sple exaples, such as the cyclc group, the dhedral group or the syetrc group A very well-nown forula gves the cycle ndex of the syetrc group S n (we adopt the notaton fro [6] here: n s j Z(S n = j j! ( One has j +2j 2 ++nj n=n = Z(S n t n = n=0 = s t, an dentty whch s of portance n varous tree countng probles (cf agan [6] The author s supported by project S96 of the Austran Scence Foundaton FWF the electronc journal of cobnatorcs 3 (2006, #N4

2 In the past, several tree countng probles related to the autoorphs groups of trees have been nvestgated We state, for nstance, the enueraton of dentty trees (see [7], and the queston of deternng the average sze of the autoorphs group n certan classes of trees (see [9, 0] Therefore, t s not surprsng that so-called cycle ndex seres or ndcatrx seres [2, 8] are of nterest n enueraton probles Gven a cobnatoral speces F, the ndcatrx seres s gven by Z F (s, s 2, = c +2c 2 +3c 3 +< f c,c 2,c 3, s c s c 2 2 s c 3 3 c c!2 c 2 c2!3 c 3 c3!, where f c,c 2,c 3, denotes the nuber of F -structures on n = c + 2c 2 + 3c 3 + ponts whch are nvarant under the acton of any (gven perutaton σ of these n ponts wth cycle type (c, c 2, (e exactly c cycles of length See for nstance [2, 6, 8] and the references theren for ore nforaton on cycle ndex seres Equvalently, t can be defned va Z F (s, s 2, = n 0 n! ( fx F [σ]x σ xσ 2 2 xσ 3 3, σ S n where fx F [σ] s the nuber of F -structures for whch the perutaton σ s an autoorphs and (σ, σ 2, s the cycle type of σ [2] In ths note, we deal wth the specal faly T of rooted trees Yet another reforulaton shows that the cycle ndex seres s also Z(Aut(T, T T where Z(Aut(T s the cycle ndex of the autoorphs group of T forula for the cycle ndex seres s due to G Labelle [8, Corollary A2]: The followng Theore The cycle ndex seres for rooted trees s gven by Z T (s, s 2, = c >0 c 2,c 3, 0 c c s c c! > c! c ( c ( j j s c,j Note that the resson resebles (, though t s soewhat longer Ths result sees to be not too well-nown, but t certanly deserves attenton In [8], Labelle proves t n a ore general settng, usng a ultdensonal verson of Lagrange s nverson forula due to Good [4] On the other hand, Constantneau and J Labelle provde a cobnatoral proof n [3] Frst of all, we wll gve a sple proof (though, of course, less general than Labelle s for ths forula, for whch only the classcal sngle-varable for of Lagrange nverson wll be necessary; then, soe edate corrolares are stated Fnally, the use of the cycle ndex seres s deonstrated by applyng the forula to the enueraton of weghted trees and -decoposable trees the electronc journal of cobnatorcs 3 (2006, #N4 2

3 2 Proof of the an theore By the recursve structure of rooted trees and the ultplcatve propertes of the cycle ndex, t s not dffcult to see that Z = Z T (s, s 2, satsfes the relaton ( Z = s Z, whch s gven, for nstance, n a paper of Robnson [2, p 344] and the boo of Bergeron et al [2, p 67] Here, Z s obtaned fro Z by replacng every s wth s Now, we prove the followng by nducton on : Z = c,,c 0 c >0 c c s c c! ( c ( c! c j =2 ( ( > d,d dc d Z,j j s c n the rng of foral power seres Then, for fnte, the coeffcent of s c sc follows at once, snce ( > d,d dc d Z doesn t contan the varables s,, s Frst note that, by Lagrange s nverson forula (cf [5, 6], we have and w = c (aw = c 0 c c x c c! a(c + a c c! x c f w = xe w Ths yelds Z = s ( Z + 2 Z = c c c s c c! c ( Z 2 whch s exactly the desred forula for = For the nducton step, we note that ( Z l = s l Z l, the electronc journal of cobnatorcs 3 (2006, #N4 3

4 and thus, by the nducton hypothess, Z = = c,,c 0 c >0 ( c,,c 0 c >0 c c s c c! =2 ( dc d d,d c c s c c! =2 ( c ( c! c j,j j s c Z + ( dc d Z > d,d< ( c ( c! c j,j ( ( j c + c j s c c c 0! j,j j,j ( ( ( c l Z l dc d Z l> > d,d< = ( c ( c c s c c! c! c j c,,c 0 c >0 =2 ( ( > Ths fnshes the nducton d,d dc d Z,j j s c j s c Corollary 2 The nuber t n = T n of rooted trees on n vertces s gven by t n = c +2c 2 +=n c >0 c c c! ( c ( c! c j j > Proof: Sply set s = s 2 = = n the dentty,j T T n Z(Aut(T = c +2c 2 +=n c >0 c c s c c! > c! c ( c ( j j s c,j As a second corollary, we obtan Cayley s forula for the nuber of rooted labeled trees Corollary 3 The nuber of rooted labeled trees on n vertces s gven by n n the electronc journal of cobnatorcs 3 (2006, #N4 4

5 Proof: Note that the coeffcent of s n n the cycle ndex of a rooted tree T on n vertces s precsely Aut(T Thus, we have But Aut(T = nn n! T T n n! Aut T s exactly the nuber of dfferent labelngs of T, whch fnshes the proof 3 Further applcatons Theore can also be appled to a general class of enueraton probles: let a set B of cobnatoral objects wth an addtve weght be gven, and let B(z be ts countng seres Now, f we want to enuerate trees on n vertces, where an eleent of B s assgned to every vertex of the tree, the countng seres s gven by c +2c 2 +=n c >0 c c B(z c c! > c! c ( c ( j j B(z c,j The coeffcent of z equals the total weght For exaple, the countng seres for rooted weghted trees on n vertces (e each vertex s assgned a postve nteger weght, cf Harary and Prns [7] s gven by W (z = c +2c 2 +=n c >0 c c c! ( c z z > c! c ( c ( ( j j,j z z c The frst few nstances are n = : W (z = n = 2: W (z = n = 3: W (z = z = z + z z2 + z 3 +, z2 = z 2 + 2z 3 + 3z 4 +, ( z 2 z3 (2+z ( z 2 ( z 2 = 2z3 + 5z 4 + 0z 5 + Fnally, we are gong to consder a new applcaton of Theore Ths exaple deals wth the decoposablty of trees: we call a tree -decoposable (a specal case of the general concept of λ-decoposablty, see [, 6] f t has a spannng forest whose coponents are all of sze It has been shown by Zelna [7] that such a decoposton, f t exsts, s always unque The specal case = 2, whch has already been nvestgated by Moon [] and Son [3, 4], corresponds to perfect atchngs Now, let D(x denote the generatng functon for the nuber of -decoposable rooted trees Snce a decoposable rooted tree s ade up fro a rooted tree on vertces (the coponent the electronc journal of cobnatorcs 3 (2006, #N4 5

6 contanng the root and collectons of -decoposable rooted trees attached to each of these vertces, we obtan the followng functonal equaton for -decoposable trees: D(x = c +2c 2 += c >0 c c E(x c c! > c! c ( c ( j j E(x c,,j where E(x = x ( D(x For = 2, we obtan ( D(x = x 2 2 D(x, gvng the nown countng seres for trees wth a perfect atchng (Sloane s A0005 [5], see also [, 3, 4]: D(x = x 2 + 2x 4 + 7x x x x 2 + For = 3, to gve a new exaple, we have ( ( D(x = 3x3 2 3 D(x + x3 2 ( D(x + D(x 2, yeldng D(x = 2x 3 + 0x x x 2 + Of course, t s possble to calculate the countng seres of -decoposable rooted trees for arbtrary n ths way The functonal equaton can also be used to obtan nforaton about the asyptotc behavor (cf [6, 6] Acnowledgent The author s hghly ndebted to an anonyous referee for provdng h wth a lot of valuable nforaton, n partcular references [2, 3, 4, 8, 2] References [] D Barth, O Baudon, and J Puech Decoposable trees: a polynoal algorth for trpodes Dscrete Appl Math, 9(3:205 26, 2002 [2] F Bergeron, G Labelle, and P Leroux Cobnatoral speces and tree-le structures, volue 67 of Encyclopeda of Matheatcs and ts Applcatons Cabrdge Unversty Press, Cabrdge, 998 the electronc journal of cobnatorcs 3 (2006, #N4 6

7 [3] I Constantneau and J Labelle Calcul cobnatore du nobre d endofonctons et d arborescences lassées fxes par une perutaton Ann Sc Math Québec, 3(2:33 38, 990 [4] I J Good Generalzatons to several varables of Lagrange s anson, wth applcatons to stochastc processes Proc Cabrdge Phlos Soc, 56: , 960 [5] I P Goulden and D M Jacson Cobnatoral enueraton A Wley-Interscence Publcaton John Wley & Sons Inc, New Yor, 983 Wley-Interscence Seres n Dscrete Matheatcs [6] F Harary and E M Paler Graphcal enueraton Acadec Press, New Yor, 973 [7] F Harary and G Prns The nuber of hoeoorphcally rreducble trees, and other speces Acta Math, 0:4 62, 959 [8] G Labelle Soe new coputatonal ethods n the theory of speces In Cobnatore énuératve (Montreal, Que, 985/Quebec, Que, 985, volue 234 of Lecture Notes n Math, pages Sprnger, Berln, 986 [9] K A McKeon The ected nuber of syetres n locally-restrcted trees I In Graph theory, cobnatorcs, and applcatons Vol 2 (Kalaazoo, MI, 988, Wley-Intersc Publ, pages Wley, New Yor, 99 [0] K A McKeon The ected nuber of syetres n locally restrcted trees II Dscrete Appl Math, 66(3: , 996 [] J W Moon The nuber of trees wth a -factor Dscrete Math, 63(:27 37, 987 [2] R W Robnson Enueraton of non-separable graphs J Cobnatoral Theory, 9: , 970 [3] R Son Trees wth a -factor: degree dstrbuton In Proceedngs of the ffteenth Southeastern conference on cobnatorcs, graph theory and coputng (Baton Rouge, La, 984, volue 45, pages 47 59, 984 [4] R Son Trees wth -factors and orented trees Dscrete Math, 88(:93 04, 99 [5] N J A Sloane The On-Lne Encyclopeda of Integer Sequences Publshed electroncally at [6] S Wagner On the nuber of decoposable trees In Proceedngs of the Fourth Colloquu on Matheatcs and Coputer Scence (Nancy 2006, pages , 2006 [7] B Zelna Parttonablty of trees Czechoslova Math J, 38(3(4:677 68, 988 the electronc journal of cobnatorcs 3 (2006, #N4 7