generating_func

# generating_func - Generating functions: Combinatorics...

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

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

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

View Full Document
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
Ask a homework question - tutors are online