A mathematical preprint reports that, when k≤n^0.25 and n is sufficiently large, every maximum-sized intersecting family in Sym(n,k) is a star. A star is a family whose members agree at one point. The result identifies both the top size and the structure of families that reach it in that stated regime.
Sym(n,k) is the paper’s name for permutations on [n] whose cycle type has exactly k cycles. Its central question is whether maximum-sized intersecting families in this setting must be stars. The unit of analysis is the mathematical family of permutations itself.
Against that maximum, the paper considers non-centred families—those not contained in a star. For k≤n^0.25, a non-centred intersecting family has asymptotic size at most (2/3+o(1)) of the largest star. In ordinary language, once the common-point structure is absent, the family cannot remain arbitrarily close to the top in the regime covered by the theorem.
The star is the benchmark
The paper writes the maximum-size bound explicitly as max{c(n−1,k), c(n−1,k−1)} for sufficiently large n with k≤n^0.25. That expression is the size of a largest star, and every intersecting family is bounded by it. The formula gives the numerical side of the result; the star classification gives its structural side.
Those statements work together. The largest family has the star form, while a family outside that form is limited to a smaller asymptotic fraction of the same benchmark. The result therefore does more than provide an upper bound on a count: it also describes the shape associated with the maximum and separates it from non-centred competitors.
A sharper result at logarithmic scale
In the narrower range k≤(ln n)^d, the preprint gives a sharper stability bound. A non-centred intersecting family has asymptotic size at most (1−1/e+o(1)) times the size of a largest star. The condition is polylogarithmic: k is controlled by a power of ln n rather than by the broader n^0.25 regime.
That constant is not just an upper-bound estimate. The paper constructs a non-centred family whose ratio to the largest star tends asymptotically to 1−1/e. The construction therefore reaches the limiting upper bound in the polylogarithmic regime, making the result asymptotically sharp.
The figures compare sizes of mathematical families inside Sym(n,k), rather than reporting an empirical measurement. The 2/3+o(1) statement and the sharper 1−1/e+o(1) statement each apply within their own range conditions, with asymptotic notation describing behavior as n becomes large.
Counting the families
The main proof uses spread approximation to extract a large centred subfamily from a sufficiently large intersecting family. That step links the behavior of a general family to the star benchmark, providing the comparison used in the argument about both the maximum and the gap below it.
The counting part reduces the number of completions of a partial permutation to c(n−|X|, k−ν(X)). The Hilton–Milner construction is then analyzed through inclusion–exclusion, and the resulting alternating sums are bounded with standard Bonferroni inequalities. These are proof tools for the stated asymptotic comparisons.
The paper denotes the Hilton–Milner-type family as Sym(n,k)[(r,s);π]∪{π}. It serves as the constructed non-centred family used to test sharpness, whose limiting ratio reaches 1−1/e in the polylogarithmic regime.
The result has a boundary
The main theorem has a clear boundary. The restriction k≤n^0.25 is attributed to limitations of the methodology, so the 2/3+o(1) conclusion is an asymptotic statement within that range. It is not presented as a claim about every possible growth rate of k.
The sharper 1−1/e+o(1) result is narrower still: it is proved for k≤(ln n)^d. The author conjectures that this sharp bound extends to k≤n^α for α∈(0,1), but that extension is a conjecture, not one of the reported bounds. The proof reaches the polylogarithmic regime, while the fractional-power range remains open.
A preprint with an open next step
The document is an arXiv preprint, version 1, dated 20 Aug 2026. It presents a proof-based asymptotic analysis, so its conclusions are tied to the stated regimes and to sufficiently large n where specified.
The acknowledgement thanks Sivaramakrishnan Sivasubramanian for suggesting the problem, and no funding source is stated. Within the supplied record, the central contribution remains mathematical: a star classification in the n^0.25 regime, a two-thirds stability bound there, and a sharp 1−1/e comparison in the polylogarithmic regime.
Put plainly, the preprint says that the largest families in the proved range are organized around one shared point. Remove that centre, and the paper’s asymptotic size bound drops to at most two-thirds of a largest star in the n^0.25 regime; in the narrower polylogarithmic range, the sharper 1−1/e bound is both proved and reached asymptotically. The outstanding question is whether the latter constant survives when k≤n^α for α∈(0,1).
Paper data and sources
Original title: Intersecting families of permutations with a fixed number of cycles
Authors: Venkata Raghu Tej Pantangi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text