Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1 x 2 2 20 x 2 1 x 3 30 x 1 x 4 24 x 5 20 x 2 x 3 fo which we ca calculate the cycle idex of the actio of S 5 o pais of vetices as,, z 6 = 1 z 10 120 1 10 z 4 1 z 3 2 15z 2 1 z 4 2 20z 1 z 3 3 30z 2 z 2 4 24z 2 5 20z 1 z 3 z 6 Now, the patte ivetoy fo a 2-coloig [p = peset, = issig] is p, p 2 2,, p 6 6 = p 10 p 9 2p 8 2 4p 7 3 6p 6 4 6p 5 5 6p 4 6 4p 3 7 2p 2 8 p 9 10 We see that thee ae 34 isoophis types ( set p = = 1) ad they ae:
2. Calculate the cycle idex fo the syetic goup S 4 actig o a) the odeed pais of distict eleets of {1,2,3,4},, z 2, z 3, z 4 = 1 24 z 12 1 6 z 2 1 z 5 2 3z 6 2 8z 4 3 6z 3 4 b) all the subsets of {1,2,3,4}., z 2, z 3, z 4 = 1 24 z 16 1 6 z 8 1 z 4 2 3z 4 1 z 6 2 8z 4 1 z 4 3 6z 2 1 z 2 z 3 4 The eueate the a) loopless digaphs, Of the 218 isoophis classes, thee is 1 with 12, 11, 1 o 0 edges, 5 with 10 o 2 edges, 13 with 9 o 3 edges, 27 with 8 o 4 edges, 38 with 7 o 5 edges ad 48 with 6 edges. b) failies of sets, Of the 3984 isoophis classes, 1 has 16 sets, 5 have 15 sets, 17 have 14 sets, 52 13 sets, 136 have 12 sets, 284 have 11 sets, 477 have 10 sets, 655 have 9 sets ad 730 have 8 sets. The ube with fewe tha 8 sets, say b sets, is the sae as the ube with 16 - b sets. o fou poits up to isoophis, by ube of edges o sets. 3. Let the cyclic goup of pie ode p with geeato g act o the set of all p-tuples of eleets of {1,...,} by the ule (x 1,..., x p )g = (x p, x 1,..., x p-1 ). By coutig obits, pove that p (od p). Let G be the cyclic goup of ode p ad X the set of p-tuples of [] that G acts o. The ube of obits of this actio is give by: O = 1 G Fix g. g G The idetity fixes evey eleet of X, but ay othe eleet of G ca oly fix those -tuples whee all the copoets ae the sae (thee ae of these). Thus, O = 1 p p p 1 = 1 p p ad sice the ube of obits ust be a itege we have p p -, that is, p od(p). 4. A double doio is a solid 2-sided chip of wood so that each side cosists of two squaes joied alog a edge. Each of the fou squaes is aked with a ube of dots betwee 0 ad k-1, iclusive. Deteie the ube of distiguishable double doioes. The goup actig o the double doio is the Klei 4-goup, Z 2 x Z 2, ad the cycle idex is ¼(x 1 4 + 3x 22 ). The ube of distiguishable double doioes is the ube of k-coloigs which is ¼(k 4 + 3k 2 ).
5. Catepillas ae tees i which the deletio of all leaves (vetices of degee 1) leaves a path. Pove that the ube of isoophis classes of -vetex catepillas is 2 4 2 / 2 2 if 3. The spie of a catepilla is the logest path i the tee with a legth of say k. This leaves -k pedat vetices to be attached to the spie i k-2 places (ca't attach at oe of the ed vetices sice that would poduce a loge spie). We thus wish to distibute -k idetical balls ito k-2 diffeet boxes. Two catepillas ae isoophic iff oe is a eflectio of the othe about a cetal vetical axis (a actio by Z 2 ), o put aothe way, if the distibutio is "palidoic", fo exaple, 3223 o 1234321. If g is the o-idetity eleet of Z 2, the by ot-buside's Lea, the ube of obits of this actio (the ube of isoophis classes) is = ½( Fix id + Fix g ). Note that k 3 fo a o-tivial catepilla. Now Fix id = k =3 k 2 k = 3 3 k =3 k = 3 =0 =2 3. The eleet g fixes the "palidoic" distibutios. To cout these, coside the cases whee k is odd ad eve. Whe k is eve so is k-2 ad a palidoic distibutio is possible oly if -k is eve, ad so ust be eve. So fo this case, let k = 2 ad = 2s ad the cotibutio is s =2 1 s s = s 2 =2 s = s 2 s 2 t=0 t =2s 2 2 =2 2. Whe k is odd, say k = 2 + 1, a ube, j, of pedats ust be attached to the cetal vetex so that -k-j is eve. We divide this ito two cases, depedig o whethe -k is eve o odd. Let -k-j = 2. If -k is eve, 0 ½(-k). If -k is odd the 1 ½(-k-1). Sice we ae foig palidoes, we eed oly distibute pedats ito the fist -1 positios (ad the epeat i evese ode fo the last -1 positios), so we have: Hece 1 = 2 = 2 2 = Fix g =2 2 2 2 2 2 =2 2 1 ad so ½( Fix id + Fix g ) = 1 2 2 3 2 2 1 =2 4 2 2 2. 1 ax 1 = =1 2 2 1 2 =2 2. 6. A Lati squae of ode is said to be syetic if a ij = a ji fo all i j. A Lati squae is idepotet if a ii = i fo all i. Pove that a syetic, idepotet Lati squae ust have odd ode. Each eleet appeas ties i the Lati Squae. A fixed eleet appeas exactly
oce o the diagoal by idepotetece, ad a eve ube of ties off the diagoal by syety. Thus, ay eleet ust appea a odd ube of ties, so (the ode) is odd. 7. Let = 2 + 1 fo soe positive itege. Pove that the x aay A = (a ij ) whee a ij (+1) x (i + j) od is a syetic, idepotet Lati squae of ode. Fo ay, by the coutivity of additio, a ij = a ji fo all i, j, so A is syetic. Fo ay i, a ii (+1)x2i (2 + 2) x i i od, so A is idepotet. Thus we eed oly show that A is a Lati Squae. Obseve that sice = 2(+1) - 1, we have that (+1, ) = 1 fo ay. Suppose that i ow i soe eleet is epeated, that is, thee ae j 1 ad j 2 so that (+1)x(i+j 1 ) (+1)x(i + j 2 ) od. Sice (+1,) = 1 this iplies j 1 j 2 od. Siilaly, if i soe colu j we had a eleet epeated, we would obtai (+1)x(i 1 + j) (+1)(i 2 + j) od ad coclude that i 1 = i 2, showig that A is a Lati Squae. 8. Let M be a Lati squae that ca be witte as a block atix X Y, whee X ad Y ae Lati Y squaes of odd ode. Pove that M has o tasvesal ad hece, o othogoal ate. Clealy M ust have eve ode, say 2 ad so, X ad Y ae of ode (odd). Suppose that T is a tasvesal of M. If thee ae b eties of T i X i the fist ows, the thee ae -b eties of T i the Y of the fist ows. This i tu eas that thee ae - (-b) = b eties of T i X of the last colus. Theefoe, thee ae exactly 2b eties of T which ae i X, but all eleets of X ust appea i T, so this cotadicts the fact that X has odd ode. 9. A [k] 2 -coveig of a gaph G is a list f 1,..., f of fuctios with f i : V(G) [k] such that fo evey edge uv of G ad evey odeed pai (,s) i [k] 2, thee is soe t such that f t (u) = ad f t (v) = s. Let g k (G) = the iiu ube of fuctios i a [k] 2 coveig of G. Pove that g k (K ) = k 2 if ad oly if thee exists a faily of -2 utually othogoal Lati squaes of ode k. Label two of the vetices of K with ad c. Fo ay [k] 2 -coveig of K, fo a table whose colus ae idexed by the vetices of K ad whose ows ae idexed by the fuctios of the [k] 2 -coveig. The eties of the table ae the values of the ow fuctio at the colu vetex. If g k (K ) = k 2, the evey pai (s,t) i [k] 2 i a uique ow i colus ad c. We will use these odeed pais to label a cell i a k x k squae. Fo each othe colu of ou table (-2 of these) we fo a squae by placig the value i a ow of the colu ito the cell labeled by the ad c eties of that ow. The coveig popety shows that the squaes foed this way ae Lati squaes of ode k ad also that ay two of these squaes ae othogoal. It is easily X
see that this pocedue is evesible ad we ca obtai a iial [k] 2 -coveig fo a set of -2 MOLS of ode k. 10. Let ad be positive iteges, <. Show that ½ is a ecessay ad sufficiet coditio fo the existece of a Lati squae of ode cotaiig a Lati subsquae of ode. Suppose that ½ ad stat with a Lati Squae of ode. Exted this Lati Squae to a Lati Rectagle by extedig the ows with ew sybols +1, +2,..., i such a way that o ew sybol appeas twice i ay ew colu (this ca be doe cyclically fo istace). As a cosequece of Hall's Maiage theoe we kow that we ca exted this x Lati Rectagle to a x Lati Squae. Clealy, this Lati Squae cotais ou oigial Lati Squae of ode as a subsquae. Now suppose that L is a x Lati Squae cotaiig a Lati Subsquae M of ode. Peute the ows ad colus of L so that the subsquae M is i the uppe lefthad coe. Noe of the eleets of M ca appea agai i eithe the ows to the ight of M o the colus below M, they ust all appea i the lowe ighthad (-) x (-) subsquae of L. So we ust have -, givig us the esult.