Bell and Stirling Numbers for Graphs
Statement
For any fixed k ≥0, the graphical Bell number sequence B(\overline{P_{n,k}}) is quasi geometric.
Context
Candidate 1 of the open problems stated in "Bell and Stirling Numbers for Graphs", extracted for the Scalable Mathematical Discovery run.
People
Attempts
No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
NEW
Problem: Reconstructed statement: for fixed , let be the -th power of the path on , with adjacent iff . Let be the number of partitions of into independent sets. A sequence is “quasi-geometric” in the paper’s sense, interpreted equivalently as satisfying a linear recurrence with constant coefficients, or having rational ordinary generating function. The conjecture asserts that
is quasi-geometric for every fixed .
Result: The conjecture is true.
Indeed, an independent block in is exactly a clique in , hence exactly a subset of of diameter at most . Thus counts set partitions of in which every block satisfies
For , only singleton blocks are allowed, so .
Fix . Let . After processing , record only the induced partition on the last vertices, encoded by their ages , where age means vertex . There are only finitely many possible such states.
Given a state, to append , either put in a new singleton block, or add it to one existing recorded block all of whose ages are at most . This condition is exactly the condition that every old member of that block lies within distance of . No unrecorded older member can occur in such a block, since that would be more than away from some recorded member of age at most , contradicting the diameter bound. Then increase all old ages by , discard age , and keep the new vertex at age .
Thus the numbers of valid partial partitions evolve by a finite transfer matrix . Therefore
for fixed finite vectors . By Cayley-Hamilton, satisfies a linear recurrence with constant coefficients. Equivalently, its ordinary generating function is rational, so it is quasi-geometric.
This also matches the small cases: gives , and gives matchings of a path, i.e. Fibonacci numbers.
Citation: No prior resolution is used here. The proof is the standard finite-state transfer-matrix/Cayley-Hamilton argument applied directly to the reconstructed conjecture.
Reviews
1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Full solution as submitted; SMD novelty classification KNOWN
PASS
The proof attacks the correct conjecture for . It correctly identifies valid blocks as subsets of of diameter at most , then gives a finite-state transfer-matrix construction for fixed by recording the induced partition on the last vertices. The key sufficiency argument—that any block eligible to receive the new vertex cannot contain unrecorded older vertices—is valid. Thus the sequence satisfies a constant-coefficient linear recurrence and has rational OGF, matching quasigeometricity. I found no fatal gap or mismatch.
Novelty assessment
KNOWN
Classification rationale: The result as accepted/proved is already recorded in OEIS in a more general form. The solution identifies with set partitions of whose blocks have . OEIS A276719 counts set partitions whose blocks are contained in an integer interval of at most elements; thus the present sequence is column . That OEIS entry explicitly states that each fixed column satisfies a linear recurrence with constant coefficients. Hence the accepted “rational OGF / constant-coefficient recurrence” resolution is not new.
A caveat: Duncan–Peele’s original “quasigeometric” definition is the stronger nearest-integer form eventually, not merely C-finiteness. The accepted solution proves the weaker interpretation, which is the one already known from OEIS.
Literature check: I checked the original Duncan–Peele paper, which proves the case and states Conjecture 4.1. Searches for “graphical Bell numbers,” “path power,” “quasigeometric,” and related bounded-span partition formulations led to OEIS A129847 for the case and, more importantly, OEIS A276719 for all bounded interval sizes. A276719 is an exact parameter-shifted version of the sequence and includes the constant-coefficient recurrence statement. ArXiv searches for graphical Bell numbers found later related papers, but not a separate new resolution of this conjecture.
Citation: Alois P. Heinz, OEIS A276719, “Number of set partitions of such that for each block the smallest integer interval containing has at most elements,” 2016. https://oeis.org/A276719
Bryce Duncan and Rhodes Peele, “Bell and Stirling Numbers for Graphs,” Journal of Integer Sequences 12 (2009), Article 09.7.1. https://cs.uwaterloo.ca/journals/JIS/VOL12/Peele/peele5.html
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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