function get_posts_for_pagination() {
    $html = '';
    $paged = ( $_GET['page'] ) ? $_GET['page'] : 1;
    $post_type = $_GET['posttype'];

    if ( empty($post_type) ) {
        return '';
    }

    if( filter_var( intval( $paged ), FILTER_VALIDATE_INT ) ) {

        $args = array(
            'post_type' => $post_type,
            'paged' => $paged,
            'posts_per_page' => 4,
            'post_status' =>'publish'
        );

        $loop = new WP_Query( $args );

        $post_count = $loop->found_posts;
        $max_num_pages = $loop->max_num_pages;

        if( $loop->have_posts() ) {
            while( $loop->have_posts() ) {
                $loop->the_post();

                $html .= 'your output'
            }
            wp_reset_query();
        }
    }

    echo $html;
    exit();
}

add_action( 'wp_ajax_pagination', 'get_posts_for_pagination' );
add_action( 'wp_ajax_nopriv_pagination', 'get_posts_for_pagination' );