bibkey: kotesovec2026a398726
authors: Vaclav Kotesovec and Alois P. Heinz
year: 2026
title: "OEIS A398726, distinct cycle size asymptotic conjecture"
doi: null
url: https://oeis.org/A398726
claim: "The sum of distinct cycle sizes over all permutations of [n] is conjectured to be asymptotic to n*n!."
strata_touched:
  - D5/S0/Asymptotics/WeightedProbability/DistinctCycleSizeLimit
license: "citation-only; OEIS source governed by https://oeis.org/LICENSE"
triage: anchor

The distinct-cycle-size conjecture

OEIS A398726 was contributed by Alois P. Heinz on August 8, 2026. Its FORMULA section attributes the following conjecture to Vaclav Kotesovec on August 14, 2026:

Conjecture: a(n) ~ n * n!.

The NAME is “Sum of the distinct cycle sizes over all permutations of [n].” For each permutation, repeated lengths contribute once, including length one when a fixed point exists. Summing those weights over every permutation gives a(n); the empty permutation contributes zero. The example a(3)=16 explicitly includes the identity’s contribution of one.

The formal source uses the parts of the permutation’s partition, which include fixed points, and converts that multiset to a finite set. It then sums the lengths over that set and over every permutation of Fin n. The conjecture’s meaning is the real ratio to n*n! tending to one as n tends to infinity. The positive-index denominator is nonzero; the value at n=0 is irrelevant to the limit.

The known entry identity a(n)=sum_{k=1}^n k*A293211(n,k) expresses the same statistic through presence of a k-cycle. A293211 and A122974 credit Dennis P. Walsh for finite cycle-avoidance/presence formulas. Those identities are background, not assumptions in the proof and not claimed new. A132961 instead counts the number of distinct lengths, without weighting by length.

Verified locator

• https://oeis.org/A398726
• Complete official text: https://oeis.org/search?q=id:A398726&fmt=text, revision 28, August 18, 2026 at 09:39:10; 1,962 bytes, SHA256 d9bdb003dcee4319284e1c8c84491a62b084e3aac4ec830699f9a31237f1c39f. NAME, the Kotesovec FORMULA line, and the n=3 EXAMPLE fix the assertion.
• https://oeis.org/A293211 and https://oeis.org/A122974, the inspected full related entries; https://oeis.org/A132961 for the different unweighted count. Walsh’s linked original note was not retrieved, so no inspection of that note or exhaustive prior-settlement claim is made.