Stacking Matryoshkas

The same split-apart nesting dolls, now with flat heads, so a smaller piece can also stand on a top half or a closed doll. Every configuration for N = 1 to 4 is drawn below.

The rules

Doll k is either closed (top on bottom) or split into its two halves.
A piece only goes inside, or on top of, a larger doll's piece.
A closed doll or an open bottom holds one item directly: a smaller closed doll or a smaller bottom (which may hold more).
An open top holds one item directly, and only a smaller top. A top never covers a doll or a bottom.
A loose top never goes inside a bottom or a closed doll.
New: the flat head of a top half or a closed doll holds one smaller item: a closed doll, a bottom or a top, with whatever it carries.
A head shut inside a closed doll or tucked under another top can't hold anything. A piece standing in an open bottom sticks out, so its head is still usable.
Position on the table doesn't matter. Only what is inside or on top of what counts.

The sequence

Further terms, N = 1…16:

2, 19, 312, 7643, 256020, 11096168, 598896401, 39139847188, 3031322144533, 273521581657006, 28358733352479982, 3339902710297719296, 442517243620250926437, 65415815270340860490360, 10712294881435175920813811, 1931132206343352778109827474

Not in the OEIS as of 2026-10-03. Without stacking the count is 2, 10, 75, 780, … (A124426, Bell(N)·Bell(N+1)). Insert the pieces from largest to smallest. Each one goes on the table or into a free slot: a flat head, an empty open bottom, the empty space inside a closed doll, or an empty top. Counting the free slots of each kind as you go gives a short recurrence. It matches a brute-force enumeration for N ≤ 5.