Course : Algebraic Combinatorics

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1 Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact way of expressig this data. We begi with the otio of ordiary geeratig fuctios. To illustrate this defiitio, we start with the example of the Fiboacci umbers. {F } =0 = {F 0, F, F 2, F 3,...} defied by F 0 =, F =, ad F = F + F 2 for 2. We defie,, 2, 3, 5, 8, 3, 2, 34,... F(X) := F 0 + F x + F 2 x2 + F 3 x3 + F 4 x = + x + 2x2 + 3x3 + 5x4 + 8x I other words, F(X) is the formal power series =0 F x. Remar. This is called a formal power series because we will cosider x to be a idetermiate variable rather tha a specific real umber. I geeral, give a sequece of umbers {a i } i=0 = {a 0, a, a 2, a 3,...}, the associated formal power series is A(X) := a x = a 0 + a x + a 2 x2 + a 3 x =0 We will shortly write dow F(X) i a compact form, but we begi with a easier example that you have already see.

2 ( )! Recall that =. For example,!( )! {( ) 8 } 8 = {, 8, 28, 56, 70, 56, 28, 8, }. =0 ( ) ( ) 8 8 I fact if > 8, (e.g. ) equals zero. Thus we ca cosider the etire ifiite 9 sequece as {( ) 8 } = {, 8, 28, 56, 70, 56, 28, 8,, 0, 0, 0,...}, =0 ad the the associated formal power series + 8x + 28x2 + 56x3 + 70x4 + 56x5 + 28x6 + 8x7 + x8 + 0x9 + 0x ca be writte compactly as ( + x) 8. {( )} Geeralizig this to ay positive iteger, has associated power series ( ) =0 ( + x), sice ( + x) = =0 x by the Biomial Theorem. This illustrates that from a formal power series, we ca recover a sequece of umbers. We call these umbers the coefficiets of the formal power series. For ( ) example, we say that is the coefficiet of x i ( + x). This is sometimes ( ) ( ) writte as ( + x) = or [x ]( + x) =. x. More complicated formal power series We ow wat to write a similar expressio for F(X) = =0 F x, where F = F + F 2 for 2 ad F 0 = F =. Notice that a x ± b x = (a ± b )x. =0 =0 =0 As a cosequece, F = F + F 2 for 2 implies F(X) = + F x + =2 F x = + F x + (F + F 2 )x =2 = + F x + F x + F 2 x =2 =2 ( ) = + F x + xf(x) F 0 x + x2 F(X).

3 Thus ad we obtai the ratioal expressio F(X)( x x2 ) = + (F F 0 )x = + 0x F(X) =. x x 2 If we loo at the Taylor series for this ratioal fuctio, we ideed obtai coefficiets that are the Fiboacci umbers. Geeratig Fuctios are also helpful for obtaiig closed formulas or asymptotic formulas. If we use partial fractio decompositio, we see that A B F(X) = +. λ x λ 2 x We ow that ( λ x)( λ 2 x) = x x 2 so Thus {λ, λ 2 } = { + 2 5, 5 }. 2 λ λ 2 λ λ 2 = ad = Exercise : Solve for A ad B ad use this to obtai a closed form expressio for F. Notice that as a cosequece we ca compute that {F + /F } = {/, 2/, 3/2, 5/3, 8/5, 3/8,...} ( ) coverges to 5 sice 5 0 as, so Aλ + + Bλ + 2 Aλ + = λ. Aλ + Bλ 2 Aλ.2 A Combiatorial Iterpretatio of the Fiboacci Numbers Give a sequece of itegers S = {s 0, s, s 2,...}, a combiatorial iterpretatio of S is a family F of objects (of various sizes) such that the umber of objects i F of size is exactly couted by s. For example, a combiatorial iterpretatio of ( ) is as the umber of subsets of a {, 2,..., } of size. A domio tilig of a rectagular regio R is a coverig of R by horizotal (-by-2) domio tiles ad vertical (2-by-) domio tiles such that every square of R is covered by exactly oe domio.

4 For example, if we let R be a 2-by-2 grid, the there are two possible domio tiligs. Either both tiles are vertical or both are horizotal. If we let R be a 2-by-3 grid, the there are three possible domio tiligs, ad a 2-by-4 grid would have five such domio tiligs. Propositio. The umber of domio tiligs of a 2-by- grid is couted by the th Fiboacci umber, F for. Proof. Let DT deote the umber of domio tiligs of the 2-by- grid. We first chec the iitial coditios. There is oe way to tile the 2-by- grid, ad there are two ways to tile the 2-by-2 grid. Thus DT = = F ad DT 2 = 2 = F 2. (Recall that F 0 =, but we do ot use this quatity i this combiatorial iterpretatio.) Domio tiligs of the 2-by- grid either loo lie a domio tilig of the 2-by ( ) grid with a vertical tile taced oto the ed, or a domio tilig of the 2-by-( 2) grid with two horizotal tiles taced oto the ed. Cosequetly, DT = DT + DT 2, the same recurrece as the F s. Exercise 2: Show that this combiatorial iterpretatio ca be rephrased as the statemet F = The umber of subsets S of {a, a 2,...,a } such that a i ad a i+ are ot both cotaied i S..3 Covolutio Product Formula I additio to addig formal power series together, we ca also multiply them. If A(X) = =0 a x ad B(X) = =0 b x, where a (resp. b ) couts the umber of objects of type A (resp. B) ad size, the A(X)B(X) = C(X) = c x where c = a b, ad has a combiatorial iterpretatio as the umber of =0 objects of size formed by taig a object of type A ad cocateatig it with a object of type B. =0

5 .4 Coectio betwee Liear Recurreces ad Ratioal Geeratig Fuctios The behavior we saw of the Fiboacci umbers ad its geeratig fuctio is a example of a more geeral theorem. Theorem. (Theorem 4.. of Eumerative Combiatorics by Richard Staley) Let α, α 2,...,α d be a fixed sequece of complex umbers, d, ad α d = 0. The followig coditios o a fuctio f : N C are equivalet: i) The geeratig fuctio F(X) equals f()x = =0 P(x) Q(x) where Q(x) = + α x + α 2 x α d x d ad P(x) is a polyomial of degree < d. ii) For all 0, f() satisfies the liear recurrece relatio f( + d) + α f( + d ) + α 2 f( + d 2) α d f() = 0. where iii) For all 0, f() = P i ()γ i i= + α x + α 2 x α d xd = ( γ i x) e i with the γ i s distict ad each P i () is a uivariate polyomial (i ) of degree less tha e i. i= Defitio. A geeratig fuctio of the form eratig fuctio. P (x) is a called a ratioal ge- Q(x)

6 MIT OpeCourseWare Algebraic Combiatorics Sprig 2009 For iformatio about citig these materials or our Terms of Use, visit: