I am sorting my posts alphabetically by Title, like so:

   <?php
      {
       $posts = get_posts($query_string . 
        '&orderby=title&order=asc&posts_per_page=-1');
      } 
      get_template_part( 'loop', 'category' );
    ?>

Let me exclude articles for example "the", "a", and "an" in the sort.

An amount be the easiest method to make this happen?

Thanks!

I'm not sure any simple method of doing that but this can be done,

For accomplishing this you have to give a custom meta field towards the publish. Title it mytitle (say).

For that new posts you add, it's easy, you need to add your modified title(getting rid of a, an, the in the title) within the mytitle custom area within the add posts page.

For old posts it's a little tricky, you need to write a php code to retrieve the game titles from the publish remove 'a','an','the' from their store using php preg_replace and combine it with the postmeta table of the wordpress database using something similar to this:

<?php //inside loop   
$query=INSERT INTO xyz_postmeta (post_id, meta_key, meta_value) VALUES ($postid, 'mytitle' $title);
$wpdb->query('$query'); ?> 

where $postid may be the publish id within the loop and $title is the modified title.

Now you must up-to-date all of the previous posts with custom mytitle area.

How to display, make use of a custom loop (not the loop incorporated within the theme).

This is how you may make a fundamental custom loop to show posts sorted so as of mytitle.

$querystr = "
   SELECT wposts.*
   FROM $wpdb->posts wposts, $wpdb->postmeta wpostmeta
   WHERE wposts.ID = wpostmeta.post_id
   AND wpostmeta.meta_key = 'mytitle'
   AND wposts.post_type = 'post'
   AND wposts.post_status = 'publish'
   ORDER BY wpostmeta.meta_value ASC
   ";

You can now execute the query at all you would like. Wordpres provides various techniques to do this. Here's a link

For instance that you can do something similar to this

$pageposts = $wpdb->get_results($querystr, OBJECT);
foreach ( $pageposts as $pagepost ) 
{
    echo $pagepost->post_title;
    //do other stuff to display content, meta etc..
}