generating_func - Generating functions: Combinatorics...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Generating functions : Combinatorics Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@ufl.edu Webpage http://www.math.ufl.edu/ squash/ 28 February, 2011 (at 15:42 ) Abstract: Examples of generating-fnc use. As usual, we will ignore the issue of series convergence. The ex- ample by Derek Ledbetter uses the M¨obius inversion formula. Nomenclature. We use Wilf’s notation from his book, Generatingfunctionology . Counting irreducible monic polynomials over a finite field This is Derek Ledbetter’s solution. Let F be a finite field; let F := # F . Henceforth All “polys” ( polynomials ) have coeffi- cients in F and are monic . 1 : ( In particular, a “poly” is not Zip. ) Let Y D denote the number of ( monic ) polys of degree– D . Note that Y D = F D , for D = 0 , 1 , 2 ... . Each poly can be written uniquely as a product of irreducibles; the constant poly 1 is the empty product. For each N Z + , let I N denote the number of irreducible 1 polys of deg– N . So I 1 = F since, for each c F , poly x + c is irreducible. 2 : Theorem. For each posint N , the number of irreducible degree– N monic polynomials is I N = 1 N X k : k •| N F k · μ ( N/k ) . 2 0 : ( Our convention for such sums is that the variable, here “ k ”, ranges only over positive divisors. ) 1 In a commutative ring, my defn of irreducible is a non–zero-divisor, non-unit which only factors trivially. The only monic degree-zero poly is 1, which is a unit in
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

Page1 / 3

generating_func - Generating functions: Combinatorics...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online