A Poem on Two Cases of Combinatorics (by aba)

Transcription

1 A Poem on Two Cases of Combinatorics (by aba) This is a rap about combinatorics I ll be feeding you theory and pushing rhetorics You can count with yo fingers, or you can count with yo mind But the latter is the way that combinatorialists have a good time. Let s break this rap up into two disjoint cases First we show graph theory, then we show enumerative faces. CASE 1: Graph Theory In 1700 in an ole Prussian town called Königsberg Seven bridges connected two islands to two banks and each other, Word! [Pause] Like this: The townspeople tried to cross one bridge at a time And come back to the place where they started and find That no bridge was crossed twice in this walk they took They couldn t solve this problem, but they knew where to look ENTER THE MOTHER OF GRAPH THEORY!! In 1700 she did give birth to all of Graph Theory, but the labor didn t hurt Cuz the mother was a brothah named Leonard Euler And cooked up Graph Theory like chicken in a broiler.

2 Instead of the Königsberg picture of land masses and water Lenny Euler said: Each land mass is just a dot And if a bridge connects two dots, then a line we got! There you got it. In a nutshell, that is Graph Theory Dots and Lines! Dots and lines, til your eyes get bleary. Euler s picture Of Königsberg You want to walk this graph crossing each edge once Euler said, Each time you pass thru a dot, you use two edges! Brilliance!!! That was the key to the bridge problem you see? A connected graph has an Euler cycle if and only if it has all vertices of even degree. It s a parity thing you see? Even or odd yo, for a graph, homie G. There s a lot more to say about graph theory But this rap would get old and yer ears would get weary Does a Hamiltonian circuit exist in that graph? That s no fair that s NP-complete! I pass. Let s move on now to Case 2 This is where you pay yo counting dues!

3 CASE 2: Enumeration So you think that you know that you know how to count? How many ways are there to distribute 8 balls into 6 boxes with the first 2 boxes having at most 4 balls If the balls are identical? What s the amount? Can you count? This problem is a horse! But can you mount? Combinations, Permutation... Arrangements, Selections, and Distributions Factorial this, Generating function Dat! Counting with these tools makes counting PHAT! The problem with the problem above Is to get your head around the constraints involved You got 8 balls and they re all alike And they go into 6 boxes, so draw something like:

4 In the first two boxes, we got a constraint They got at most 4 balls, in here 5 or mo AIN T So we split the problem up into disjoint situations Then add the totals up cuz that the additive principle of enumeration If 0 balls are in the first two boxes, then there is 11 choose 8 ways to put the rest in the four remaining boxes. There s 2 choose 1 ways to put one ball in the first two boxes, and 10 choose 7 ways to put the other 7 in the four remaining boxes. There s 3 choose 2 ways to put two balls in the first two boxes, and 9 choose 6 ways to put the other 6 in the four remaining boxes. There s 4 choose 3 ways to put three balls in the first two boxes, and 8 choose 5 ways to put the other 5 in the four remaining boxes. There s 5 choose 4 ways to put four balls in the first two boxes, and 7 choose 4 ways to put the other 4 in the four remaining boxes. We add this all up cuz these cases are disjoint! We get a final answer like this: That s the point! The answer is 1056.

5 So that s the spirit of what we call combinatorial reasoning It s a careful balance of mathematical seasoning. You can t take this class unless you ve had calculus But where it come in? Sometimes it s a plus. But the rhyme is getting long and I can t explain Taylor series jack the house in generating function ways! And also set theory provides a nice program To set up Inclusion-Exclusion using Sven s diaphragm? Oops I think I meant to say something else Venn diagrams are like Santa s enumeration elves. So that concludes the case of enumerative combinatorics This rap has gone long in endless loops of rhetorics So let me just leave you with a question on Will s and my mind: Will Richard was a student in my Combinatorics class. During a research project with Jen Taback, he came to me to ask about a certain product of generating functions. This led us to the following function above.