WordPress & Ajax

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This link to the WordPress Codex should help. It demonstrates a few different ways to POST using AJAX in a plugin. As for re-arranging your listed elements, this can be done a number of ways.

I would recommend running a search for rearranging elements using javascript. The jQuery function .insertBefore() and .insertAfter() can be used to drag/drop elements. For example, the following code can be used to move an element above the previous element:

$("p").click(function() {
    $(this).insertBefore($(this).prev()); 
});

or even something like this just to manually move a paragraph above the other:

<p id="1"></p>
<p id="2"></p>
<!--javascript-->
<script type="text/javascript">
    $('#2').insertBefore('#1');
</script>

It’s hard to help out much more without any further information.