# How to express a chromatic polynomial as a falling factorial

#### Solarmew

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 $$\displaystyle p(P_{3},x) = x(x-1)^{2}$$
and $$\displaystyle p(C_{4},x) = (x-1)^{4} + (x-1)$$
and i dunno how to express that as a falling factorial :|

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

and $$\displaystyle (x^{(3)}+x^{(2)})\vee(x^{(4)}+2x^{(3)}+x^{(2)})\not=x(x-1)(x-2)(x-3)(x^3- 12x^2 + 50x-71)$$ :cry:
edit2: holy crap, it totally does = :shock:
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? :shock:

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#### Solarmew

Re: How to express a chromatic polynomial as a falling facto

nm, pls delete post

#### Solarmew

Re: How to express a chromatic polynomial as a falling facto

nm, pls delete post