(k) x n. n! tk = x a. (i) x p p! ti ) ( q 0. i 0. k A (i) n p

Transcription

1 Math 880 Bigraded Classes & Stirlig Cycle Numbers Fall 206 Bigraded classes. Followig Flajolet-Sedgewic Ch. III, we defie a bigraded class A to be a set of combiatorial objects a A with two measures of magitude, a primary measure a = called simply the size, ad a secodary measure a = called the weight or parameter, or a statistic o A. Usually we cosider labeled bigraded classes Ã, i which each object a à has = a atoms havig all the labels, 2,...,, as well as the weight a urelated to the labels: i particular, a permutatio of the labels should ot chage either a or a. We have the coutig umbers A () = #{a à with a =, a = }. The classic example is the labeled class P = 0 S comprisig all permutatios w S, with size w = ad weight w = cyc() = umber of cycles of w. I this case the coutig umbers are the Stirlig cycle umbers: [ () = P = #{w S cyc(w) = }. We tae the bivariate geeratig fuctio: Ã(x, t) = A () x! t = x a a! t a. a à All the costructios available for labeled graded classes exted to the bigraded case. I particular, we edow the labeled product à B with the additive weight fuctio (a S, b T ) = a + b, where a S meas a à with its atoms relabeled by the set S. We have the Labeled Bigraded Product Priiciple: for C = à B, the bivariate geeratig fuctio is C(x, t) = Ã(x, t) B(x, t). The proof is very similar to the sigle-graded case: Ã(x, t) B(x, t) = ( p 0 = = i 0 A p (i) x p p! ti ) ( q 0 p=0 i=0 ( ) A (i) p j 0 p B ( i) p C () x! t = C(x, t). B q (j) x q q! tj ) x! t The secod iequality uses the chage of idex variables = p + q ad = i + j, so that ( ) p! = p!q!. The third equality is because each elemet (a () S, b T ) C correspods to the choice of S [, a Ã(i) ( i) p, b B p. Similarly, the costructios Seq j, Seq, Set j, Set, Cyc j, Cyc ca be performed o bigraded classes, ad give the same formulas for the bivariate geeratig fuctios. Coutig cycles i permutatios. The class of labeled cycles of legth is Cyc [, where [ is the sigle-elemet labeled graded class, with bivariate expoetial geeratig fuctio! x! t = x t, sice a cycle is the same as a permutatio of [ up to rotatio equivalece. Here x! idicates the size, while t = t idicates that a sigle cycle has weight. Allowig cycles of ay legth gives geeratig fuctio x t = t log( x ). Realizig ay permutatio as a set of labeled cycles gives the bivariate geeratig fuctio of

2 P = 0 S = Set(Cyc [): P (x, t) = [ x! t = 0 ( ) ( ) t log(! x ) = exp t log( x ) = ( x) t. We ca get several iterestig specializatios of this bivariate fuctio. First, taig t = gives us the sigle-variable expoetial geeratig fuctio: ( ) P (x) = exp log( x ) = x. Ideed, we ca realize permutatios either as sets of cycles or as labeled sequeces: P = Set(Cyc [) = Seq( [); ad the above computatio is the geeratig fuctio versio. Next, fixig ad taig the t coefficiet gives the sigle-variable geeratig fuctio: P () (x) = 0 [ x! =! ( t log( x ) ). This does ot give a explicit formula for the Stirlig cycle umbers, but it allows complex aalytic methods to give the asymptotic approximatio: [ ( )! ( )! (log ) as. This meas that for large, the fractio of -permutatios which have cycles is very close to (log ) ( )!. Fially, fixig ad taig the coefficiet of x! gives the geeratig fuctio P (t) = t. The Taylor Coefficiet Formula gives: [ = P (t) = d dx P (x, t) = x=0 ( ) d dx ( x) x=0 t = t(t+)(t+2) (t+ ) = t. That is: = [ t = t. Substitutig t for t ad factorig out sigs chages risig powers to fallig: [ ( ) t = t(t ) (t +) = t. = Compare this with our formula ivolvig Stirlig partitio umbers: = { } t = t. These formulas mea that ( ) [ ad { } are the chage-of-basis coefficiets betwee the usual power basis, t, t 2, t 3,... for the polyomials i t, ad the fallig power basis, t, t 2, t 3,.... Hece, if for ay N we defie lower-triagular matrices M = ( ( ) [ ) N,= Bijective proof. The right side t couts all fuctios f : [ [t for t N, each of which ca be factored ito a surjective fuctio, surj(, ) =! { } ( ; ad a choice of image, t ) =! t. 2

3 ad M 2 = ({ N }),=, the these are iverse matrices: M M 2 = I = (N N) idetity matrix. This is equivalet to the formula: [ ( ) j j j 0 { } j = { if = 0 if. Similarly, M 2 M = I is equivalet to: { ( ) j j j 0 } [ j = { if = 0 if. These formulas ca also be proved by the Ivolutio Priciple, the first usig the same /00 Ivolutio used for t = [ j= ( ) j j t j o p. 5. The secod ca be proved usig the Loely/Crowded Ivolutio. The left side couts permuted set partitios: a uordered partitio S S j = [, umbered so that mi(s ) < < mi(s j ), alog with a permutatio w S j, ad give the sig ( ) j. The ivolutio will defie a ew permuted set partitio (S, w ). Tae the smallest a [ such that, for a S i, the cycle of sets S i S w(i) S w(w(i)) cotais at least two elemets. If S i = {a} is a sigleto, the joi it with the ext set o its cycle: S i = S i S w(i) ad w (i) = w(w(i)). If S i is ot a sigleto, split it ito two sets alog the same cycle: S i = {a} ad S w (i) = S i {a}, with w (w (i)) = w(i). This chages the sig ( ) j by icremetig/decremetig j while leavig, fixed. If there is o such a, the this is the uique fixed poit cosistig of all sigleto sets ad w = id. { } [ j j 0 ( )j j the right side. I the siged cout, I pairs up ad cacels all terms except the fixed poit, which gives Cycle formula. We give aother proof of: t = [ ( ) t. = For a whole umber value t N, the left side t ca be iterpreted as the umber of: ijective fuctios f : [ [t proper t-colorigs of the complete graph K sequeces (a,..., a ) of a i [t with a i a j for i < j vectors i the hyperplae complemet F t i<j H ij, where F t is a fiite field, ad H ij is the hyperplae havig the i, j coordiates equal. We ca obtai the right side from the fourth iterpretatio, by usig Mobius iversio (geeralized PIE) o the lattice L(B) of subspaces V F t geerated by the braid arragemet B = i<j H ij. We give this the partial order of reverse iclusio: U V meas U V, so that the miimal elemet is the etire space ˆ0 = F t. This poset is isomorphic 3

4 to Π, the set partitios S = {S,..., S } with S S = [, ordered by refiemet. Note that uder this isomorphism, dim(v ) is equal to l(s) =. Usig the fuctios f, g : L(B) Z give by: f(v ) = #V = t dim(v ), g(v ) = #(V H V H) where the uio rus over all hyperplaes H = H ij ot cotaiig V. The: f(u) = V U g(v ) g(u) = V U µ(u, V )f(v ) = V U µ(u, V )t dim(w ), where µ(w, U) is the Mobius fuctio of L(B) defied by µ(u, U) = ad U V W µ(u, V ) = 0 for U < W. I particular: t = #(F t H ij) = g(ˆ0) = µ(ˆ0, V ) t dim(v ) = µ(ˆ0, S) t l(s). i<j S Π V L(B) The above expressio is called the characteristic polyomial of the subspace arragemet: i fact, the chromatic polyomial of ay graph is equal to the characteristic polyomial of the correspodig graphical hyperplae arragemet. Taig the t term of the above expressio, correspodig to the maximal elemet ˆ = V = F t (,..., ), we fid that µ(ˆ0, ˆ) = µ(f t, 0) = ( ) ( )!, ad from the product poset structure of the iterval [ˆ0, S for S = {S,..., S } with l(s) = ad i = #S i, we easily fid that µ(ˆ0, S ) = ( ) i ( i )!. Therefore: t = ( l(s) ( ) l(s) S Π i= ) ( i )! t l(s). Now, we ca costruct ay permutatio by partitioig [ ito subsets of size,...,, ad puttig the elemets of each bloc ito a i -cycle i oe of ( i )! ways. Thus, for a give = l(s), the above expressio couts all permutatios with of [ with cycles, ad we have: [ t = ( ) t. = We obtai yet aother formula for t from the secod ad third iterpretatios above. A cojuctio of coditios a i = a j ca be regarded as a set of pairs {i, j} i the powerset ( ) ( [ 2, a Boolea poset. Ordiary PIE correspodig to the poset [ ) 2 gives: t = ( ) e(g) t c(g), G K where G rus over all graphs o labeled vertices, with e(g) edges ad c(g) coected compoets. Comparig this to the previous formula, each pair {i, j} ca be cosidered as a relatio i j, ad a set of pairs geerates a equivalece relatio correspodig to a set partitio i Π. Note that each S Π correspods to c c sets i ( ) [ 2, where cj is the umber of coected graphs o j labeled vertices, so there are much fewer terms i 4

5 the Π formula, ad much fewer still i the =,..., formula. Bijective proof of cycle formulas. We first prove the positive formula: t = = [ t. The left side couts permutatio partitios: that is, a ordered set partitio of [ ito sets (S,, S ), where S i may be empty, alog with (w,..., w ), where w i S Si is a permutatio of the set S i. (Oe may picture this as a arragemet of distict flags o t flagpoles.) The right side couts pairs (w, f) w S F w, where F = {fuctios f : [ [t} ad F w meas the fuctios ivariat uder w, i.e. f is costat o each cycle of w, so that F w = t cyc(w). There is a easy bijectio betwee these objects which proves the formula. Give a permutatio partitio (S,..., S ; w,..., w ), let f(j) = i for j S i, ad let w = w w. Coversely, give (w, f), let S i = f (i), ad let w i be the permutatio w restricted to S i. We may also cosider this as a example of Polya s Method (or Burside s Lemma) coutig orbits of group actios. Dividig both sides by! gives: (( )) t =! = [ t. The left side couts multisets with objects of t ids, which is the quotiet of F uder the atural actio of S. Now Burside s Lemma gives: F S = F w, S w S where F w is the set of fuctios ivariat uder w. This traslates directly to the two sides of the multiset formula. Fially, we give a ivolutio proof of the siged formula t = [ ( ) t. = Now the left side couts ijective fuctios E = {f : [ [t}, while the right side is the siged cout of w S F w, where we defie sg(w, f) = ( ), where w has cycles. Now we defie a sig-reversig ivolutio I : F w w S w S F w with fixed-poit set ( w S F w ) I = (id, E), which will prove the formula. For (w, f) w S F w with f ijective, let I(w, f) = (w, f); sice f is costat o all cycles of w, it must have oly -cycles, so w = e ad (w, f) (id, E). If f ot ijective, suppose f(a) = f(b) for miimal a, b [; the defie I(w, f) = (w(ab), f), multiplyig w by the traspositio (ab), so that w(ab) = (cd)w for c = w(a), d = w(b). This reverses sig, 5

6 sice if a, b lie i the same cycle of w, the w(ab) cuts this cycle ito two; if a, b lie o differet cycles, the w(ab) jois these cycles ito oe. Thus, I will cacel all o-ijective (w, f) w S F w o the right side of the formula, leavig oly the ijective (id, E) o the left side. We call this the /00 Ivolutio because it splits a figure-eight cycle ito two cycles, ad vice versa: it is useful because it toggles every reasoable defiitio of sig for w, icremetig/decremetig both the umber of cycles ad the umber of traspositios. This argumet may also be see as a example of coutig orbits, this time with sigs. Dividig by! gives: ( ) t =! [ ( ) = The left side couts -elemet subsets of [t, which is the quotiet of E uder the free actio of S. Also E = (id, E) = ( w S F w ) I, so by the Ivolutio Priciple: E S t. = S ( F w ) I = ( ) cyc(w) F w, S w S w S which clearly traslates to the siged cycle formula. 6