A392962 - OEIS

%I #40 Mar 31 2026 01:19:27

%S 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

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

%C 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.

%C 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.

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

%H Igor Blokhin, <a href="https://github.com/IgorBlokhin/Graph-Theory">Graph theory</a> (Python repository)

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

%Y Cf. A000055, A391694.

%K nonn,hard,more

%O 1,4

%A _Igor Blokhin_, Mar 27 2026

%E a(22)-a(25) from _Sean A. Irvine_, Mar 31 2026