I am a bit confused by various descriptions of this plant but below is some thing that relates to its appearance.
The singular appearance of the Jack-in-the-Pulpit's mace-like cluster of red berries at the top of an otherwise bare stem has inspired a plethora of folk names, each with a parochial etymology. Many of the names were originally from various European species of the arum family with similar taxonomies that were later applied to the native A. triphyllum by early settlers. Arums are characterized by the flower-bearing spadix which is at the center of the plant surrounded by the leafy sheath-like spathe. The red berry cluster is the seed bearing remnant of the spadix after the spathe has fallen away.
Very lovely, I saw them in the forest here as well, but looking at your shot I think I should take a closer view the next time. Thanks for the explantion