1 Generating functions for balls in boxes

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1 Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways beig to write Geeratig fuctio = a 0 f 0 x + a f x + a f x + + a f + where the fuctios f k x are called idicator fuctios. Ordiary geeratig fuctios use f k x = x k as idicator fuctios so the the ordiary geeratig fuctio for a 0, a, a,..., a,..., is a 0 +a x+a x + +a x +. I usually deote this fuctio by Ax. If I had a sequece of b k, I d use Bx, etc. A closed form for a geeratig fuctio is a simple expressio for the ifiite or fiite sum. For example, if we use the sequece of biomial coefficiets, 0,,...,, the the geeratig fuctio is 0 + x + x + + x, ad by the Biomial Theorem, this has closed form + x. The Cauchy Covolutio of two sequeces: If Ax = a 0 + a x + a x + + a x +, ad Bx = b 0 + b x + b x + + b x +, the their product is AxBx = a 0 b 0 + a 0 b + a b 0 x + a 0 b + a b + a b 0 x +. Here, the coefficiet of x i AxBx will be a 0 b + a b + a b + + a b 0. Geeratig fuctios for balls i boxes Oe of the uses for ordiary geeratig fuctios is for problems ivolvig balls i boxes, or iteger solutios to equatios. As metioed i class this basically follows from the Cauchy covolutio. That is to say, suppose you have the problem of askig how may ways balls ca be put ito a box. The aswer is oe way. The resultig sequece is for = 0, for =,,,..., with geeratig fuctio + x + x + x + = x. Suppose we have two boxes, but for some reaso, the umber of ways to put balls ito the boxes are a for the first box ad b for the secod box. How may ways ca we put balls ito the two boxes? We could have 0 balls i the first, i the secod, or oe ball i the first, i the secod, ad so o. The cout is a 0 b + a b + a b + + a b 0, a Cauchy covolutio, so the geeratig fuctio is the product of the idividual geeratig fuctios. I the simplest case, =, ad by iductio, this exteds to ay umber of boxes: If you have m boxes, thig of the first m of them as beig oe complicated x x x box with its geeratig fuctio ad the m th box as aother. Example: Fid the umber of ways to put balls ito 5 boxes if the last three boxes ca hold at most balls.

2 Solutio: The geeratig fuctio is the product +x+x +x + +x+x +x + +x+x +x +x+x +x +x+x +x = x x x 4 x x 4 x x 4 x = x4 x 5. Next, we extract the coefficiet of from this. The umerator expads as x 4 + x 8 x, ad the deomiator is the geeratig fuctio for 5+. By the Cauchy covolutio, the aswer we seek is , or with costat deomiator, Example: Fid the geeratig fuctio for the umber of oegative iteger solutios to x + x + x + x 4 =. Solutio: This is the same as puttig balls ito 4 boxes, where the secod ad third boxes eed a eve umber of balls ad the last box eeds a multiple of three. The geeratig fuctio is + x + x + x + + x + x x + x x + x 6 + = x x x x = x x x. Geeratig fuctios for series calculatios Here, I am iterested i two problems:. Give a sequece a 0, a, a,..., a,..., with correspodig geeratig fuctio Ax = a 0 + a x + a x + + a x +, how do we fid a closed form for Ax?. Give closed expressio for Ax, ca we fid a ice formula for a 0 + a + a + + a? Some tools for this: If Ax = a 0 + a x + a x + + a x +, the the geeratig fuctio for the sequece a 0, a 0 + a,..., a 0 + a + + a,... is Ax. Aother useful x trick is that if Ax is the geeratig fuctio for a 0, a, a,..., a,..., the d Ax is the geeratig fuctio for a, a, a,..., a,..., ad x d Ax is the geeratig fuctio for 0, a, a, a,..., a,.... Example Fid a closed form for the sequece, 9, 4,..., 4 + +,..., ad use this to fid a formula for Page

3 Solutio: To fid a closed sum for the sequece with a = 4 + +, we ca break the problem ito three subproblems: Fid closed forms for + x + x + x + + x +, x + 6x + 9x + + x +, 4x + 6x + 6x x +. The first is easy sice it is a geometric series summig to. The secod ca be x foud via the x d operator: +x+x + = so x x+6x +9x + +x + = x d = x. For the last term, we deed two applicatios of the x d operator. Note x x that each applicatio gets harder. We have 4x + 6x + 6x x + = x d Thus, our geeratig fuctio is = x d = x d x d x d 4 x 4x x 4 + 4x + 4x + + 4x + = x 4 x 8x x x 4 = x + x 4x + 4x + x + x + =. x x x 4x + 4x x. Now that we have this geeratig fuctio, we divide by x to get the geeratig fuctio for the sum, + x + x GF = x 4 We wat the coefficiet of x i this geeratig fuctio, ad this is = , which simplifies, if we like, to We could have worked with x + x 4x + 4x + rather tha the combied x x expressio. I that case, whe extractig the coefficiet of x we would have had Page

4 Solutio: Here is a alterative approach. Sice x = m+ m + k m x k, the geeratig fuctio for the sequece i of m+ m is. We try to write x m as a combiatio of biomial coefficiets. We might be able to guess that = a + b + c = a + b + + c. The trickiest way I kow to solve for a, b, c is to substitute values i for. If we set = we get a =. If we substitute =, we have = a b so b = 9. Fially, at = 0, = a + b + c so c = 8. This meas the geeratig fuctio for the sequece is k + k + + k + 4k x k = x k 9 x k + 8 x k = x 9 x + 8 x. We divide by x to get the geeratig fuctio for the sum, ad the extract the coefficiet of x from that expressio. The result is = Let s do a slightly more complicated example. Example: Fid a formula for the sum Solutio: I fact, this is just a geometric series, ad should be doe that way, but we ca use geeratig fuctios to do this as well. The sum looks like a Cauchy covolutio. I fact, it is the covolutio of the sequeces { } ad { }. The geeratig fuctios for these sequeces are ad x x, respectively, so the geeratig fuctio for the covolutio is the product, x x. We eed the coefficiet of x i this expressio. The way to do this is to use partial fractios: x x = a x + b x for some choice of a ad b. The usual way to fid a ad b is to clear deomiators ad substitute key values for x. Here, we have = a x + b x. Settig x = gives b = ad settig x = gives a = 6. Thus, our geeratig fuctio ca be writte as 5 5 Page 4

5 6 5 x 5 x. The coefficiet of x i this is This could be expresses as = As a check, this was a geometric series with first term ad ratio 6 be 6 = = so the sum should A cute way to partially check aswers I our example with we got several aswers for the sum: , , , Are these all the same? Here is a way to check based o two facts: First, if we wat to check that two polyomials are equal, say px = qx, ad the maximal degree of the polyomials is k, the it is eough to check that the polyomials agree at at least k + poits. Secod, oe ca view a biomial coefficiet such as + as a polyomial i. I fact, it is a cubic. To check that the formulas match, we just eed to check that they agree at 4 poits. We ca use k = 0 whe < k i such checks. The easiest values of to check are probably = 0,,,. I additio, I will ote that the actual sum has values,, 5, ad 8 for = 0,,,. I will let you check that takes o values,, 5, 6 8 for = 0,,,, givig weight to the correctess of the formula. Here, I will check the biomial coefficiet versios. I the first case we have = 0 : + + = =, 4 = : + + = =, Page 5

6 5 4 = : + + = = 5, = : + + = = 8. I ll let you check the secod biomial coefficiet formula. I the third case we have = 0 : = =, 4 = : = =, 4 5 = : = = 5, = : = = 8. Fially, here is the hard example from Moday s class: Fid the umber of ways to put balls ito 4 boxes if the first box must have a eve umber of balls, the secod box a odd umber, the third box is oempty, ad there is o restrictio o the fourth x box. The geeratig fuctio was. Oe way was to multiply umerator ad x 4 + x deomiator by + x ad try to extract the coefficiet of x from x + x. We have x 4 x x 4 + x = x + x + x 4 x 4 k + = x + x + x 4 x k k + k + k + = x k+ + x k+ + x k+4 Whe is eve, oly the first ad third terms cotribute. Whe is odd, oly the secod term cotributes. The coefficiet of x will be m+ + m+, if = m, m+, if = m +. The other approach, which I oly partially did i class was to write the geeratig fuctio x as ad use partial fractios. I obviously made mistakes i class because Maple x 4 +x tells me that x x 4 + x = 4 x 4 4 x 6 x x. Page 6

7 Usig this, the coefficiet of x should be This secod aswer ca be simplified quite a bit. First, + + = + secod, + = +, allowig the aswer to be writte ad This The sequece is fairly simple: 0, -, 0, -, 0, -,.... So the aswer is a biomial coefficiet with a correctio for the odd terms. Page 7