Repeated Sizes: Stacking

N dolls where several can share a size. Dolls of the same size are identical, and only the order of the sizes matters. Pieces nest, and smaller pieces can also stand on flat heads. Every arrangement for N = 1 to 4 is drawn below, grouped by size set.

The rules

Each of the N dolls has a size, and sizes can repeat. Only their order matters: sizes 1, 1, 3 are the same set as 1, 1, 2 or 2, 2, 3. So a size set is a list of how many dolls share each size, smallest first, and there are 2^N−1 of them.
Dolls of the same size are identical: swapping them changes nothing, and any top fits any bottom of its size.
Each doll is either closed or split into its two halves.
A piece only goes inside or on top of a strictly larger piece. Equal sizes never nest or stack.
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.
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. The count adds up the arrangements over every size set.

The sequence

Further terms, N = 1…13:

2, 22, 415, 12160, 493998, 26174898, 1735381834, 139755623586, 13367574533502, 1492053442567399, 191592227514173127, 27972153968339834054, 4597659910322013131297

Not in the OEIS as of 2026-10-03. The size set with all sizes distinct gives the stacking sequence 2, 19, 312, 7643, …; a single shared size gives N + 1 (just how many dolls are closed). The terms come from Burnside's lemma: average, over the ways of permuting equal pieces, the number of arrangements each permutation leaves unchanged. Those counts reduce to the slot-counting recurrence of the distinct-size case. A brute-force enumeration of every arrangement agrees for N ≤ 4, and a Pólya multiset count agrees for N ≤ 6.