Mathematical connections in combinatorial enumeration

Combinatorics Complex Analysis Algebra and Geometry Conclusion

Mathematical connections in combinatorialenumeration

Timothy DeVries [email protected]

University of Pennsylvania

Bi-College Math ColloquiumHaverford, 2010

Connections in Combinatorics DeVries


Outline

1 Enumerative CombinatoricsWhat We Count and How We Count ItThe Generating Function

2 Applying Complex AnalysisCauchy’s Formula and Coefficient ExtractionPemantle-Wilson Analysis, Part I

3 Enter Algebra, Geometry and TopologyThe Singular VarietyPemantle-Wilson Analysis, Part II



What We Count and How We Count It

Outline







What is Enumerative Combinatorics?

Enumerative Combinatorics is the mathematics of counting —typically counting discrete mathematical objects.

A class of such objects is usually partitioned by a setproperties, and the task is to count the sizes of these partitions.

These properties are typically each indexed by naturalnumbers.




ExamplesThe Fibonacci Numbers

You start with a pair of rabbits, born at the start of month 1. Apair of rabbits mates at the beginning of every month, startingwhen they are one month old. One month after mating, eachpregnant rabbit gives birth to a new rabbit pair. Rabbits neverdie. How many pairs of rabbits fn are there at the start of monthn?

Answer: The Fibonacci numbers:

f1 = 1, f2 = 1, f3 = 2, f4 = 3, f5 = 5, f6 = 8, . . .

fn = fn−1 + fn−2 for n > 2




Examples, cotd.The Binomial Coefficients

On an r × s grid, how manyways br ,s are there to travelfrom the lower left corner ofthe grid to the upper rightcorner, stepping only up andright.Answer: The binomialcoefficients.

br ,s =

(r + s

r

)Figure: The

(42

)= 6 ways of

navigating a 2× 2 board.




Examples, cotd.The Delannoy Numbers

On an r × s grid, how many ways dr ,s are there to travel fromthe lower left corner of the grid to the upper right corner,stepping only up, right or diagonally up and right.

Answer: The Delannoy numbers.

Figure: An example path on a 3× 5 grid. Note: d3,5 = 231.




What Do We Mean By Counting?

There are many different things we may mean by counting,including• Obtaining a closed form counting function.• Obtaining a recursive counting function.• Obtaining an asymptotic counting function.

It is the third of these possibilities with which we will beconcerned.




Asymptotic Counting

DefinitionTwo functions f and g are said to be asymptotic, and we writef ∼ g, if

limn→∞

f (n)

g(n)= 1.

Thus, to count asymptotically means to come up with anapproximating formula whose relative error goes to zero.




An Example of Asymptotics

Example

For the (diagonal) binomial coefficients, we have(2nn

)∼ 4n√πn

A question you might ask yourself: what earthly right does πhave showing up in that formula!?



The Generating Function

Outline







An Overview

Generating functions are the main tool we will use to countasymptotically. So what are they?

“A generating function is a clothesline on which wehang up a sequence of numbers for display”.

– Herbert Wilf

To a sequence of numbers, we associate an algebraic object —a formal power series — in which the sequence is embedded.




The Definition

DefinitionThe (ordinary) generating function associated to a (bivariate)sequence ar ,s of numbers is the formal power series

F (x , y) =∞∑

r ,s=0

ar ,sx r ys.

Idea: Sometimes this defines a holomorphic function (near theorigin). Thus we may be able to use complex analysis to studythe coefficients.




Example: The Binomial Coefficients

Denoting the generating function by

F (x , y) =∑r ,s

br ,sx r ys,

the recurrence relation br ,s = br−1,s + br ,s−1 (for r , s ≥ 1) leadsto the relation

F (x , y)(1− x − y) = 1,

and so we haveF (x , y) =

11− x − y

.



Cauchy’s Formula and Coefficient Extraction

Outline







Cauchy’s Formula

TheoremLet U ⊆ C be an open set on which a function F is holomorphic,and let D ⊆ U be a closed disc. Then for x in the interior of D,

F (x) =1

2πi

∫∂D

F (z)

z − xdz.

Note: this is actually just a form of Stokes’ Theorem (thefundamental theorem of calculus):∫

Ddω =

∫∂Dω,

where the “differential” is actually a Dirac delta function.




Differentiating Cauchy’s Formula

From Cauchy’s formula we have

dn

dxn F (x) =dn

dxn1

2πi

∫∂D

F (z)

z − xdz

Differentiating under the integral sign, this yields

F (n)(x) =n!

2πi

∫∂D

F (z)

(z − x)n+1 dz

F (n)(x)

n!=

12πi

∫∂D

F (z)

(z − x)n+1 dz

But the left hand side of this equation is just the formula for thenth coefficient in the Taylor expansion of F at x .




Coefficient Formula (one variable)

TheoremLet F (z) =

∑∞n=0 anzn be a function holomorphic in a

neighborhood of the origin. Then we have

an =1

2πi

∫∂D

F (z)

zn+1 dz,

where ∂D is a sufficiently small circle about the origin.

Computation of the coefficients reduces to computation of anintegral.




Coefficient Formula (two variables)

TheoremLet F (x , y) =

∑∞r ,s=0 ar ,sx r ys be a function holomorphic in a

neighborhood of the origin. Then we have

ar ,s =1

(2πi)2

∫∫T

F (x , y)

x r+1ys+1 dx dy

where T is the product of sufficiently small circles about theorigin in the x and y planes.

This follows by iterated integration.

Note: similar formulas exist for any number of variables.



Pemantle-Wilson Analysis, Part I

Outline







The Situation

Pemantle-Wilson deals only with the case where the generatingfunction is rational. Specifically, we have:

• A sequence ar ,s that we wish to count• The sequence’s generating function

F (x , y) =∑∞

r ,s=0 ar ,sx r ys.

• We assume that F is holomorphic near the origin, where ittakes the form P/Q for some polynomials P and Q.

For simplification, we will only examine the case whenr = s = n, with n→∞.




Motivating Idea

We must analyze the integral∫∫T

Pxn+1yn+1Q

dx dy

as n→∞.

We see that the asymptotic magnitude of the integrand isgoverned by |x | and |y |.

Idea: Push the domain of integration T out toward infinity,where the integrand’s magnitude is smaller.




An Outline in One Variable

• Push contour towardinfinity (doesn’t changeintegral, thanks toStokes’ Theorem).

• Push contour aroundsingularities.

• Integral far from origin isasymptotically negligible.

• Integral near singularitiesreduces to the value of aresidue function at thepoint.




















































































































































What Happens in Two Variables?

In more than one variable, the outline remains essentiallyunchanged except for the last step.

In the final step we must compute the integral of a residuefunction over a cycle on the singular set (reduces problem byone dimension).

Thus we must understand the geometric structure of thesingular set, specifically where Q = 0.



The Singular Variety

Outline







Algebraic Geometry

Algebraic Geometry is the study of geometric problems asapproached using an algebraic framework.

Of prime interest to classical algebraic geometry: Thecorrespondence between a set S of polynomials and thevanishing set V of S, called an algebraic variety.

We are interested in the variety VQ of the vanishing set (in C2)of a single polynomial Q(x , y).




An Example: Bicolored SupertreesAs an example, take

Q(x , y) = x5y2 + 2x2y − 2x3y + 4y + x − 2

What does VQ look like?

Q is quadratic in y , so we can solve

y =−x2 + x3 − 2 +

√x4 + 4x2 − 4x3 + 4x5

For most values of x we get two values of y . As parameterizedby x , VQ basically looks like two sheets that connect together insome complicated way.




An Example, cotd.Visualization

Can show that VQ is a smoothsurface, but it is a two (real)dimensional surface in a four(real) dimensional space.How do we picture that?

Idea: throw out one of thedimensions! (ignore =y ).

Figure: A depiction of VQ inthree dimensions.



















































































































Pemantle-Wilson Analysis, Part II

Outline







The Last Step

What remains: to compute the (one-dimensional) integral of aresidue function along a cycle on VQ.

The asymptotic magnitude of the integrand is still governed bythe magnitudes of x and y .

Idea (again): Manipulate the contour of integration to minimizethe size of the integrand. We define a height function:

h(x , y) = − ln |x | − ln |y |

and attempt to minimize the maximum height along the contourof integration.




The Last Step, cotd.

The idea: flow contour down to lower heights (minimize themaximum height on the contour)

Where might this get stuck? Saddle points of the heightfunction!

This is very good! Near saddle points, we can apply the methodof steepest descent (asymptotic formulae).




An Example: Bicolored Supertrees

The variety VQ; now coloredby the height function. Redindicates high points andviolet indicates low points.

Figure: Manipulating the contourof integration, bicoloredsupertree example














































Homology and Morse Theory

We are hitting upon two very important topics in topology.

• Homology Theory: We replace the contour of integrationby one which is in some sense equivalent. Homologycaptures this notion of equivalence.

• Morse Theory: We want to find an equivalent contourwhose height is maximized at a saddle point. MorseTheory deals directly with calculating homology, and howhomology is affected by critical points of a height function.




Finishing our Computation

With the height along the contour of integration maximized atsaddle points, the integral locally breaks into:

• saddle-point integrals near these points, and• negligible pieces away from these points.

Finally, we apply saddle point integration formulas to obtainasymptotics.




Saddle Point Integration

A simple example theorem:

TheoremFor any ε > 0, we have:∫ ε

−εe−

nx22 dx ∼

√2πn

Note: theorems exist for integrands of the more general form

A(x)enh(x)

where h′(0) = 0 and h(x) is maximized at x = 0.



Closing Points

Future Research:• How might we automate this task (two variable case)?• How can we perform this analysis in three and more

variables (the topology gets much more complicated at thecritical values of the height function)?

Connections in Mathematics:“Solving a problem simply means representing it so asto make the solution transparent.”

– Herbert Simon



Further Reading

Pemantle, R. and Wilson, M.Twenty combinatorial examples of asymptotics derivedfrom multivariate generating functions.SIAM Review, vol. 50, pages 199-272.

DeVries, T.A case study in bivariate singularity analysis.http://www.math.upenn.edu/~tdevries/ ; to appear inCONM, 2010.


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Mathematical connections in combinatorial enumeration