Repeated Sizes: Nesting

N dolls where several can share a size. Dolls of the same size are identical, and only the order of the sizes matters. Pieces only nest. 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 a strictly larger piece. Equal sizes never nest.
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.
Position on the table doesn't matter. The count adds up the arrangements over every size set.

The sequence

Further terms, N = 1…18:

2, 13, 117, 1485, 24508, 505381, 12612491, 372322941, 12770803390, 501779797457, 22322464263069, 1113465408407831, 61762804180641774, 3782719976340625333, 254226434258030713275, 18647276568817135935691, 1485581867072414191122855, 127996356073773681330787566

Not in the OEIS as of 2026-10-03. The size set with all sizes distinct gives A124426, Bell(N)·Bell(N+1); 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.