Your safest bet is not to have AdSense on your landing page (but have it everywhere else if you want)
Unless they've changed something, you are allowed to put it on your landing page, technically, but most people unfortunately aren't expert enough at crafting a good landing page and they don't do it right. So they get banned.
The important thing to consider is what the end goal is here. Are you an ecommerce site? then your landing page should be completely focused on getting that conversion, and an ad there is pretty much an admission of failure.
Is the goal just to get traffic to the site? That's perfectly legitimate, but you're making a gamble that the cost of bringing them in is going to be significantly lower than the cost of sending them away if they see an ad on your site and click on it. Personally, I gave it up when I realized I was spending money to get people on my site only to have them leave again - d'oh.
So first you think about what YOU want, and then you have to think about what Google wants. Obviously they would love you to run AdWords to your site. But the AdSense group isn't usually thrilled about driving paid traffic to their ads, because that often isn't good converting traffic for their advertisers. So it might turn some eyeballs in your direction from G.
(Note - don't send traffic to your search, that will get you a horrible QS. Probably even sending it to the directory is not a great idea. Make a special landing page just for PPC, outlining the benefits of each)