Multiplying two generating functions
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1 Generating Functions Multiplying two generating functions Let A(x) = a k x k and B(x) = b k x k. k 0 k 0 What is the coefficient of x k in A(x)B(x)?
2 Generating Functions Multiplying two generating functions Let A(x) = a k x k and B(x) = b k x k. k 0 k 0 What is the coefficient of x k in A(x)B(x)? When expanding the product A(x)B(x) we multiply terms a i x i in A by terms b j x j in B. This product contributes to the coefficient of x k in AB only when.
3 Generating Functions Multiplying two generating functions Let A(x) = a k x k and B(x) = b k x k. k 0 k 0 What is the coefficient of x k in A(x)B(x)? When expanding the product A(x)B(x) we multiply terms a i x i in A by terms b j x j in B. This product contributes to the coefficient of x k in AB only when. Therefore, A(x)B(x) = k 0 ( k i=0 a i b k i ) x k
4 Generating Functions Multiplying two generating functions Let A(x) = a k x k and B(x) = b k x k. k 0 k 0 What is the coefficient of x k in A(x)B(x)? When expanding the product A(x)B(x) we multiply terms a i x i in A by terms b j x j in B. This product contributes to the coefficient of x k in AB only when. Therefore, A(x)B(x) = k 0 ( k i=0 a i b k i ) x k An interpretation of this theorem: If a k counts all sets of size k of type S, and b k counts all sets of size k of type T, then [x k ] ( A(x)B(x) ) counts all pairs of sets (S, T ) where the total number of elements in both sets is k.
5 Generating Functions Multiplying two generating functions Example. What is the coefficient of x 7 in x 3 (1+x) 4 (1 2x)?
6 Generating Functions Multiplying two generating functions Example. What is the coefficient of x 7 in x 3 (1+x) 4 (1 2x)? A special case of the above: (A(x)) 2 = k 0 ( i 1 +i 2 =k a i1 a i2 ) x k
7 Generating Functions Multiplying two generating functions Example. What is the coefficient of x 7 in x 3 (1+x) 4 (1 2x)? A special case of the above: (A(x)) 2 = k 0 ( i 1 +i 2 =k Similar to above, the coefficient of x k of (A(x)) n is: (A(x)) n = ( ) a i1 a i2 a in x k k 0 i 1 +i 2 + +i n=k a i1 a i2 ) x k
8 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially?
9 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =
10 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =1+G (x)+g (x)2 + G(x) 3 +.
11 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =1+G (x)+g (x)2 + G(x) 3 +. This is an infinite sum of (likely infinite) power series. The constant term h 0 of H(x) only makes sense if
12 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =1+G (x)+g (x)2 + G(x) 3 +. This is an infinite sum of (likely infinite) power series. The constant term h 0 of H(x) only makes sense if g 0 =0.
13 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =1+G (x)+g (x)2 + G(x) 3 +. This is an infinite sum of (likely infinite) power series. The constant term h 0 of H(x) only makes sense if g 0 =0. This implies that x n divides G (x) n. Hence, there are at most n 1 summands which contain x n 1. We conclude that the infinite sum makes sense.
14 Compositions of Generating Functions 113 Compositions of Generating Functions Question: Let F (x) = n 0 f nx n and G (x) = n 0 g nx n. What does H(x) =F ( G (x) ) represent combinatorially? Investigate F (x) =1/(1 x). F (G(x)) = 1 1 G(x) =1+G (x)+g (x)2 + G(x) 3 +. This is an infinite sum of (likely infinite) power series. The constant term h 0 of H(x) only makes sense if g 0 =0. This implies that x n divides G (x) n. Hence, there are at most n 1 summands which contain x n 1. We conclude that the infinite sum makes sense. For a general composition with g 0 =0, F (G(x)) = f n G(x) n = f 0 + f 1 G (x)+f 2 G (x) 2 + f 3 G (x) 3 +. n 0
15 Compositions of Generating Functions 114 Compositions. of. Generating Functions. 1 Interpreting 1 G(x) =1+G (x)+g(x)2 + G(x) 3 + : Recall: If G(x) represents the number of ways to build a certain structure on an n-element set, then G (x) k represents the number of ways to split the n-element set into an list of k pieces and build a structure on each piece.
16 Compositions of Generating Functions 114 Compositions. of. Generating Functions. 1 Interpreting 1 G(x) =1+G (x)+g(x)2 + G(x) 3 + : Recall: If G(x) represents the number of ways to build a certain structure on an n-element set, then G (x) k represents the number of ways to split the n-element set into an list of k pieces and build a structure on each piece. Conclusion: As long as g 0 =0,then1+G (x)+g(x) 2 + G (x) 3 + represents all ways to break down an n-element set into any number of (ordered) pieces and build a structure on each piece.
17 Compositions of Generating Functions 114 Compositions. of. Generating Functions. 1 Interpreting 1 G(x) =1+G (x)+g(x)2 + G(x) 3 + : Recall: If G(x) represents the number of ways to build a certain structure on an n-element set, then G (x) k represents the number of ways to split the n-element set into an list of k pieces and build a structure on each piece. Conclusion: As long as g 0 =0,then1+G (x)+g(x) 2 + G (x) 3 + represents all ways to break down an n-element set into any number of (ordered) pieces and build a structure on each piece. A partition: p 1 + p p k = n with p 1 p 2 p k. A composition: c 1 + c c k = n with no restrictions.
18 Compositions of Generating Functions 114 Compositions. of. Generating Functions. 1 Interpreting 1 G(x) =1+G (x)+g(x)2 + G(x) 3 + : Recall: If G(x) represents the number of ways to build a certain structure on an n-element set, then G (x) k represents the number of ways to split the n-element set into an list of k pieces and build a structure on each piece. Conclusion: As long as g 0 =0,then1+G (x)+g(x) 2 + G (x) 3 + represents all ways to break down an n-element set into any number of (ordered) pieces and build a structure on each piece. A partition: p 1 + p p k = n with p 1 p 2 p k. A composition: c 1 + c c k = n with no restrictions. Think: A composition of generating functions equals a composition. of. generating. functions.
19 Compositions of Generating Functions 115 An Example, Compositions Example. How many compositions of n are there? Solution. We break down n into pieces of size k. What is the structure on each piece? For a piece of size k > 0, there is one option it s a piece of size k. Therefore G(x) =x + x 2 + x 3 + x 4 + =
20 Compositions of Generating Functions 115 An Example, Compositions Example. How many compositions of n are there? Solution. We break down n into pieces of size k. What is the structure on each piece? For a piece of size k > 0, there is one option it s a piece of size k. Therefore G(x) =x + x 2 + x 3 + x 4 + = The generating function for compositions is H(x) = 1 1 G(x) =
21 Compositions of Generating Functions 115 An Example, Compositions Example. How many compositions of n are there? Solution. We break down n into pieces of size k. What is the structure on each piece? For a piece of size k > 0, there is one option it s a piece of size k. Therefore G(x) =x + x 2 + x 3 + x 4 + = The generating function for compositions is H(x) = 1 1 G(x) = So the number of compositions of n is
22 Compositions of Generating Functions 116 A Composition Example Example. How many ways are there to take a group of n soldiers, break them into non-empty platoons and assign each platoon a leader?
23 Compositions of Generating Functions 116 A Composition Example Example. How many ways are there to take a group of n soldiers, break them into non-empty platoons and assign each platoon a leader? Solution. We break down the n soldiers set into groups of size k. What is the structure on each group? For a group of size k > 0, there are options.
24 Compositions of Generating Functions 116 A Composition Example Example. How many ways are there to take a group of n soldiers, break them into non-empty platoons and assign each platoon a leader? Solution. We break down the n soldiers set into groups of size k. What is the structure on each group? For a group of size k > 0, there are options. Therefore G(x) =
25 Compositions of Generating Functions 116 A Composition Example Example. How many ways are there to take a group of n soldiers, break them into non-empty platoons and assign each platoon a leader? Solution. We break down the n soldiers set into groups of size k. What is the structure on each group? For a group of size k > 0, there are options. Therefore G(x) = And the generating function for such a military breakdown is 1 1 2x + x2 H(x) = = 1 G(x) 1 3x + x 2
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