A337114 - OEIS

%I #22 Feb 25 2021 21:27:19

%S 1,1,1,2,3,6,9,15,19,32,36,56,70,103,122,175,210,298,349,486,569,773,

%T 912,1237

%N Number of distinct node-partitions of n-vertex trees.

%H Gordon Hamilton, <a href="https://mathpickle.com/project/treefolk-tribes-symmetry-sorting-sequence">Treefolk Tribes</a>

%e There are three node-partitions of 5-vertex trees. 1) The star graph has a unique central element four indistinguishable leaves. This corresponds to the 1-4 partition. 2) 5-vertices in a line has a unique central vertex. Both neighboring vertices are indistinguishable. The two leaves are indistinguishable. This corresponds to the 1,2,2 partition. 3) The remaining 5-vertex tree corresponds to the 1,1,1,2 partition.

%e Among the 12-vertex trees, there are many which share a node partition. For example there are four which share the node partition: 1,1,1,1,3,5 and seventy nine that share the node partition: 1,1,1,1,1,1,1,1,1,1,2. Of the seventy-seven partitions of 12 there are 21 that have no associated tree 77-a(12) = 77-56 = 21.

%K nonn,more

%O 1,4

%A _Gordon Hamilton_, Aug 16 2020

%E a(13)-a(24) from _Bert Dobbelaere_, Aug 25 2020