I created a new Inventory of type InventoryType.CARTOGRAPHY with 

Inventory inventory = Bukkit.createInventory(player, InventoryType.CARTOGRAPHY);

and showed it to the player using

player.openInventory(inventory);

There are several things that do not work:

• I cannot set items using inventory.setItem()
• If I drag-click items into slot 0 or 1 they just disappear and get removed from the cursor (See attachment)
• If I add a filled map into slot 0 (by clicking on it, not via code, as this doesn't work) no image is shown.

If I place a cartography table on the ground and use it's Inventory things work as usual.

Sourcecode:

public final class CustomInventoryTest extends JavaPlugin implements Listener {
    @Override
    public void onEnable() {
        this.getServer().getPluginManager().registerEvents(this, this);
    }

    @EventHandler
    public void on(PlayerJoinEvent event) {
        Player player = event.getPlayer();
        Inventory inventory = Bukkit.createInventory(player, InventoryType.CARTOGRAPHY);
        inventory.setItem(1, new ItemStack(Material.PAPER));
        player.openInventory(inventory);
    }
}

Things I tried:

• Open the Inventory with a delay
• Adding the Item after opening the Inventory
• Adding the Item with a delay
• Adding Items (also with different Materials) to other slots
• Creating the Inventory without an InventoryHolder