My Math Forum How to express a chromatic polynomial as a falling factorial

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December 15th, 2012, 02:27 PM   #1
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How to express a chromatic polynomial as a falling factorial

How do i express a chromatic polynomial as a falling factorial function?
[attachment=0:szbeqan8]Capture.JPG[/attachment:szbeqan8]
I understand the join-product, but it seems that in order to apply it I'd need to convert the chromatic polynomials into falling factorial functions
like $p(P_{3},x)= x(x-1)^{2}$
and $p(C_{4},x)= (x-1)^{4} + (x-1)$
and i dunno how to express that as a falling factorial

edit: i think i kinda get it .... but i'm getting:
$p(P_{3},x)= x^3-2x^2+x = x^{(3)}+x^{(2)}$
$p(C_{4},x)= x^4-4x^3+6x^2-3x = x^{(4)}+2x^{(3)}+x^{(2)}$

and $(x^{(3)}+x^{(2)})\vee(x^{(4)}+2x^{(3)}+x^{(2)})\no t=x(x-1)(x-2)(x-3)(x^3- 12x^2 + 50x-71)$
edit2: holy crap, it totally does =
thanks everybody XD this is why i love this forum ... even when noone's responding it's still helpful XD
my only question left is how do i do this without actually drawing out all the picture and using chromatic reduction?
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 December 15th, 2012, 02:48 PM #2 Senior Member     Joined: Feb 2010 Posts: 199 Thanks: 0 Re: How to express a chromatic polynomial as a falling facto nm, pls delete post
 December 15th, 2012, 03:46 PM #3 Senior Member     Joined: Feb 2010 Posts: 199 Thanks: 0 Re: How to express a chromatic polynomial as a falling facto nm, pls delete post

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