One of the things I worked on at the clinic last summer was what I've come to call the almost-not-posting trot. I'm a fan of posting trot in general - it's a great way to allow the horse to warm up, whether you're riding English or Western, and if you want to move at a trot that's more than a shuffle/jog, posting can be more comfortable.
But posting can cause its own set of issues. Your seat is away from the horse, so you lose the connection there. And there are lots of ways to post that create braces and blocks in your body and therefore interfere with your horse's motion and softness. (Of course, a sitting trot where you're stiff or braced, and your back and hips don't move freely, can be pretty darn uncomfortable for you and the horse both.)
Posting often creates a lot of movement in your body, and is frequently done from a brace - in legs, hips or back - and often involves muscular effort which creates its own bracing. None of that helps your horse move or your connection with the horse.
The thing about almost-not-posting is that you can't do it if you're braced or blocking. The main idea is to use the lift from the horse's hip to create a very slight rise in your hip - just barely there. The closest analogy is posting the trot when riding bareback or with no stirrups, but often that's done with a lot of pinching or muscular effort - the almost-not-posting trot should be effortless.
If your heels are jammed down, creating bracing and tension in your legs, you won't be able to do the almost-not-posting trot. If your knees or thighs are pinching, same thing. If your back or hips are stiff and not moving well, or if you're carrying an artificial, "mannered", arch in your back (often seen, incorrectly, in hunter riding), or if you've got your butt tucked under and you're driving with your seat (frequently seen in dressage and Western riding), the almost-not-posting trot can't happen. If you're pushing up from your feet, you're probably squeezing at the same time, which creates a brace on every rise. If your balance is off because you're looking down or your head and chin are dropped, not happening. If your contact with your horse's mouth is braced, or your hands, leg and seat aren't independent, no way.
The almost-not-posting trot requires a neutral position, and relaxation and softness in your body. This, by the way, is independent of what your horse is doing - the horse can be excited or trying to brace - the almost-not-posting trot can actually help the horse "come to you" in terms of softness and relaxation. I've found that, when I can do it properly - it certainly isn't there yet for me all the time and I still have to think about it to get it - my connection with the horse becomes much more consistent, and I'm much more "in" rather than "on" the horse.
My almost-not-posting trot work grew directly out of letting go in my lower back, starting at the walk - I'd had a big defensive brace going on there for years due to some earlier back problems. Feeling the horse's hips and barrel move at the walk, and following/allowing with seat, back and legs (no pushing and maintaining a neutral, soft position) is very close to the feeling of the almost-not-posting trot, and good practice for it.
When a horse trots, and we're posting, we often think of it as rising with the outside front leg. Instead, think about how the horse's hind legs and hips are moving. As the outside front leg moves forward, the hind leg on the same side is pushing and that hip is rising - that's the propulsion for the almost-not-posting trot. Feel the horse's hips rising and falling at the walk, and allow yourself to move with it - even exaggerate the motion as a form of practice. Then do the same thing at sitting trot, being particularly attentive to the alternating upwards push of the hind legs in turn. (This - feeling the motion of the horse's hips - is also is the key to being able to post on the correct diagonal without having to look at the horse's shoulders.) Now, just allow one of the horse's hips to create some very minimal motion in your hips and lower back - just the slightest lift of your hip, and very, very slight rotation of your pelvis. Your seat should only just slightly come out of the saddle - maybe as much as an inch but not much more - and then follow the horse's hip back down. If someone's looking at you, they may have trouble even telling that you're posting - that's the almost-not-posting trot.
I've found it to be a very powerful way to ride the posting trot, and I'm working on making it the only way I post, so my connection can be more consistent. A ways yet to go on that, but as always, my horses are working hard to teach me . . .