Only display authors who have posts

Question

I am using the code below to display all authors and their posts (in a custom post type called "publication"). This works, however it also lists users who do not have posts. How can I adapt this code to only show authors who have posts?

Thank you!

    <h1>Articles by Author</h1>

    <?php 
    $allUsers = get_users('orderby=title&order=ASC&post_type=publication');
    $users = array();
    // Remove subscribers from the list as they won't write any articles
    foreach($allUsers as $currentUser)
    {
        if(!in_array( 'subscriber', $currentUser->roles ))
        {
            $users[] = $currentUser;
        }
    }
    ?>

    <?php foreach($users as $user) { ?>

    <div class="authorInfo">
    <h2 class="authorName"><?php echo $user->display_name; ?></h2>

     <?php query_posts( array( 'post_type' => 'publication', 'showposts' => -1, 'author'=>$user->ID ) ); while (have_posts()) : the_post(); ?>
            <?php the_content(); ?>
            <?php endwhile; ?>
    </div>

    <?php } ?>

UPDATE: Using the function from the answer below and a few things I read last night, here's what I have now (still not working as requested above, but wanted to give you my most recent code):

<?php 

// count number of Articles (CPT) written by a user
function count_user_posts_by_type( $userid, $post_type = 'publication' ) {
    global $wpdb;
    $where = get_posts_by_author_sql( $post_type, true, $userid );
    $count = $wpdb->get_var( "SELECT COUNT(*) FROM $wpdb->posts $where" );
    return $count;


$post_count = count_user_posts_by_type($userid, $post_type);
$allUsers = get_users('orderby=title&order=ASC&post_type=publication&role=author');
$users = array();

// CONDITIONAL -- IS THIS CORRECT?
if ($post_count > 0 ) {

// Remove subscribers from the list as they won't write any articles
foreach($allUsers as $currentUser)
{
    if(!in_array( 'subscriber', $currentUser->roles ))
    {
        $users[] = $currentUser;
    }
}
foreach($users as $user) { ?>


<h2 class="authorName"><?php echo $user->display_name; ?></h2>

 <?php query_posts( array( 'post_type' => 'publication', 'showposts' => -1, 'author'=>$user->ID, 'orderby' => 'date', 'order' => 'DESC' ) ); 
 while (have_posts()) : the_post(); ?>
        <?php the_content(); ?>
        <?php endwhile; ?>


<?php } } ?>

Answer 1

The default version of the WordPress function provides the count of the default post type 'POST' as stated below:

<?php
    $post_count = count_user_posts($userid);
?>

For retrieving the post count based on a post type a custom function as below is required:

<?php
    function count_user_posts_by_type( $userid, $post_type = 'post' ) {

        global $wpdb;

        $where = get_posts_by_author_sql( $post_type, true, $userid );

        $count = $wpdb->get_var( "SELECT COUNT(*) FROM $wpdb->posts $where" );

        return $count;
    }
?>

Usage of the above function:

<?php
    $post_count = count_user_posts_by_type($userid, $post_type);
?>

Now as you have got the post count wrap your code in a conditional loop.

How to use in your function:

<?php
// count number of Articles (CPT) written by a user
function count_user_posts_by_type( $userid, $post_type = 'publication' ) {
    global $wpdb;
    $where = get_posts_by_author_sql( $post_type, true, $userid );
    $count = $wpdb->get_var( "SELECT COUNT(*) FROM $wpdb->posts $where" );
    return $count;
}
$allUsers = get_users('orderby=title&order=ASC&post_type=publication');
$users = array();
// Remove subscribers from the list as they won't write any articles
foreach($allUsers as $currentUser)
{
    if(!in_array( 'subscriber', $currentUser->roles ))
    {
        $users[] = $currentUser;
    }
}

foreach($users as $user) { 

    $post_count = count_user_posts_by_type($user->id);

        if ($post_count > 0 { ?>

        <div class="authorInfo">
        <h2 class="authorName"><?php echo $user->display_name; ?></h2>

         <?php query_posts( array( 'post_type' => 'publication', 'showposts' => -1, 'author'=>$user->ID ) );
            while (have_posts()) : the_post();
                the_content(); 
            endwhile; ?>
        </div>

    <?php }
} 
?>