A392962 Number of automorphism group orders that occur for exactly one unlabeled tree with n vertices.

1, 1, 1, 2, 1, 3, 6, 6, 7, 5, 9, 15, 18, 19, 23, 26, 23, 34, 39, 42, 45, 48, 60, 65, 79

Offset

1,4

Comments

Let T(n) be the set of all unlabeled trees with n vertices. For each tree T in T(n), let Aut(T) be its automorphism group and |Aut(T)| its order.

For each integer m, define f(n, m) = number of trees T in T(n) such that |Aut(T)| = m. Then a(n) is the number of integers m such that f(n, m) = 1. Equivalently, a(n) counts those automorphism group orders that occur exactly once among all unlabeled trees with n vertices.

Compare with A391694, which counts all distinct orders without regard to multiplicity.

Links

Table of n, a(n) for n=1..25.

Igor Blokhin, Graph theory (Python repository)

Example

n = 5: There are three unlabeled trees with automorphism group orders 24, 2, 2. Only 24 occurs exactly once. Hence a(5) = 1.

Crossrefs

Cf. A000055, A391694.

Sequence in context: A118287 A391611 A024930 * A121966 A349980 A021472

Adjacent sequences: A392959 A392960 A392961 * A392963 A392964 A392965

Keyword

nonn,hard,more

Author

Igor Blokhin, Mar 27 2026

Extensions

a(22)-a(25) from Sean A. Irvine, Mar 31 2026

Status

approved