Most likely a .32-20 Hand Ejector Model of 1905, 4th Change, manufactured from 1915 to 1940, with S/Ns ranging from 65701 to 144684.
+1, two other quick notes about the gun...
1) Some .32-20 ammo is meant for Winchester rifles and is loaded hotter than what your S&W can handle. AFAIK all modern ammo is safe, but if you find some older ammo, check the label carefully for a disclaimer saying something like "SAFE FOR REVOLVERS" or "RIFLES ONLY". Using "Rifle" ammo in your .32-20HE could result in a gun-destroying kB!
If you handload, the load books usually also separate .32-20 ammo into "Rifle" (too hot!) and "Pistol" (just right) sections. Choose your ammo wisely!
2) .32-20HE "Target" models with an adjustable rear sight are quite rare and have become quite valuable in recent years. Many were produced on a special-order basis, so some have been found with some highly unusual factory options such as McGivern gold-bead front sights and extra-wide checkered(!) triggers. If you have a .32-20 Target, we'd be eager to see pictures of it.