How to propagate property change notifications of objects within collections

EDIT: Overhauling this answer to be more specific to the question since the tags show you already knew about INotifyPropertyChanged.

You need to implement INotifyPropertyChanged in class A and in class S. Make it so objectA can only be set through a property that will raise the PropertyChanged event on S whenever a property is changed in A. Example:

public class A : INotifyPropertyChanged
{
    private string name;
    public string Name
    {
        get { return name; }
        set { name = value; OnPropertyChanged("Name"); }
    }

    private int count;

    public int Count
    {
        get { return count; } 
        set { count = value; OnPropertyChanged("Count"); }
    }

    public event PropertyChangedEventHandler PropertyChanged;
    protected void OnPropertyChanged(string propertyName)
    {
        if (PropertyChanged != null) PropertyChanged(this, new PropertyChangedEventArgs(propertyName));
    }
}

... and class S...

public class S : INotifyPropertyChanged
{
    private string name, ID;
    private A objectA;

    public A ObjectA
    {
        get { return objectA; }
        set
        {
            var old = objectA;
            objectA = value;

            // Remove the event subscription from the old instance.
            if (old != null) old.PropertyChanged -= objectA_PropertyChanged;

            // Add the event subscription to the new instance.
            if (objectA != null) objectA.PropertyChanged += objectA_PropertyChanged;

            OnPropertyChanged("ObjectA");
        }
    }

    void objectA_PropertyChanged(object sender, PropertyChangedEventArgs e)
    {
        // Propagate the change to any listeners. Prefix with ObjectA so listeners can tell the difference.
        OnPropertyChanged("ObjectA." + e.PropertyName);
    }

    public event PropertyChangedEventHandler PropertyChanged;
    protected void OnPropertyChanged(string propertyName)
    {
        if (PropertyChanged != null) PropertyChanged(this, new PropertyChangedEventArgs(propertyName));
    }
}

For class R, use ObservableCollection<S> instead of List<S>, and subscribe to its CollectionChanged event, and monitor when objects are added or removed to listOfObjectS. When they are added, subscribe to S's PropertyChanged events. Then updated R's view. Example:

public class R
{
    protected string name;
    protected System.Collections.ObjectModel.ObservableCollection<S> ListOfObjectS { get; private set; }

    public R()
    {
        // Use ObservableCollection instead.
        ListOfObjectS = new ObservableCollection<S>();

        // Subscribe to all changes to the collection.
        ListOfObjectS.CollectionChanged += listOfObjectS_CollectionChanged;
    }

    void listOfObjectS_CollectionChanged(object sender, System.Collections.Specialized.NotifyCollectionChangedEventArgs e)
    {
        if (e.Action == NotifyCollectionChangedAction.Remove)
        {
            // When items are removed, unsubscribe from property change notifications.
            var oldItems = (e.OldItems ?? new INotifyPropertyChanged[0]).OfType<INotifyPropertyChanged>();
            foreach (var item in oldItems)
                item.PropertyChanged -= item_PropertyChanged;
        }

        // When item(s) are added, subscribe to property notifications.
        if (e.Action == NotifyCollectionChangedAction.Add)
        {
            var newItems = (e.NewItems ?? new INotifyPropertyChanged[0]).OfType<INotifyPropertyChanged>();
            foreach (var item in newItems)
                item.PropertyChanged += item_PropertyChanged;
        }

        // NOTE: I'm not handling NotifyCollectionChangedAction.Reset.
        // You'll want to look into when this event is raised and handle it
        // in a special fashion.
    }

    void item_PropertyChanged(object sender, PropertyChangedEventArgs e)
    {
        if (e.PropertyName.StartsWith("ObjectA."))
        {
            // Refresh any dependent views, forms, controls, whatever...
        }
    }
}