u/johnBassoon

Hiii r/math rejected me so im posting hereee!! im not really a math person so i used some ai to organize this,, i checked everything but call it out if anything sounds like bullshit😂

"
A decomposition tree of a positive integer n is a rooted tree in which each node is labeled by a positive integer. The root is labeled n. For every non-leaf node with label k, the labels of its children [can be arbitrarily many, not just 2!!] are positive integers whose sum equals k. All labels appearing anywhere in the tree must be >>distinct(unique)<<.
The depth of the root is defined as 0. The tree has depth at least m if every leaf node occurs at depth ≥ m.
Define a sequence of sets LEAF_n recursively as follows:
- LEAF_0 ⊆ Z+ is an initial set of allowed labels.
- For n ≥ 1, LEAF_n is the subset of LEAF_(n-1) consisting of those integers that can appear as the root label of a decomposition tree of depth at least n, where each node at distance m from the root belongs to LEAF_(n-m).
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Sooo about the image: the first tree is invalid because 3 appears twice. The second tree is NOT depth 2 because not all leaf nodes have depth ≥ 2. The yellow line just means "this doesn’t matter" 🥲
s(n) just means the smallest element of LEAF_n and the image shows s(n) when LEAF_0 = Z+.
s(0) = 1 trivially.
s(1) = 3 = (1 + 2) obviously.
s(2) = 11 = (4 + 7) = ((1 + 3) + (2 + 5)), and you can manually check that it's the only possible decomposition (disregarding symmetric forms and shittt). Brute force searching found that s(3) = 39 = (16 + 23) = ((6 + 10) + (11 + 12)) = (((1 + 5) + (2 + 8)) + ((4 + 7) + (3 + 9))) and idk if this is the only decomposition,, but did you notice something cool?
When T_n denotes the n-th triangular number:

s(0) = 1 = T_1
s(1) = 3 = T_2
s(2) = 11 = T_4 + T_1
s(3) = 39 = T_8 + T_2
(!) s(4) = T_16 + T_4 + T_1????

I have so many questions!! Can you answer'em all!? When LEAF_0 = Z+, is any integer ≥ s(n) an element of LEAF_(n)? Does s(n) always have only one valid decomposition tree[resolved, no]? Is the triangular number pattern correct? Is it possible for a decomposition tree of s(n) to include a node with three children? Am I hallucinating all of this!!?!???

Y concept

a/A