The p-order method got a lot of attention in the solution of Askey's 1986 problem 6514 in the Math Monthly to show that $$\frac{(3m + 3n)!(3n)!(2m)!(2n)!}{(2m + 3n) !(m + 2n) !(m + n)!m!n!n!}$$ is always an integer. 

It was conjectured that this is the constant term of $$\left(  \left( 2-{\frac {y}{x}}-{\frac {x}{y}} \right)  \left( 2-{
\frac {y}{z}}-{\frac {z}{y}} \right)  \left( 2-{\frac {z}{x}}-{\frac {
x}{z}} \right)  \right) ^{m} \left(  \left( 2-{\frac {yz}{{x}^{2}}}-{
\frac {{x}^{2}}{yz}} \right)  \left( 2-{\frac {yx}{{z}^{2}}}-{\frac {{
z}^{2}}{yx}} \right)  \left( 2-{\frac {xz}{{y}^{2}}}-{\frac {{y}^{2}}{
xz}} \right)  \right) ^{n}$$ (and hence an integer). This was established in  A Proof of the $G_2 $ Case of Macdonald's Root System-Dyson Conjecture by Doron Zeilberger, SIAM J. Math. Anal. 18, 880 (1987), DOI:10.1137/0518065 . So this is certainly not a p-order proof. However I don't know that there are constant term identities for these other ratios. The article does cite (a special case of) a theorem of Morris  showing that the following expression is a constant term and hence an integer: $$\frac{(a+b+2c)!(a+b+c)!(a+b)!(2c)!(3c)!}{
(a+2c)!(b+2c)!(a+c)!(b+c)!a!b!c!c!} $$