A mathematical preprint describes a new family of permutation group polynomials over finite fields of odd characteristic, together with an explicit companion polynomial and exact counts for the constructions it studies. The paper also states that the new family is inequivalent to every known family considered in the manuscript. The result concerns algebraic constructions and permutation groups; no empirical sample or dataset is reported.
In the central construction, q is p cubed, with p an odd prime. The paper says the group generated by its two permutations is a subgroup of the group of permutations on q elements and has order q. Every non-identity element has no fixed point in the finite field F_q. Put simply, each nontrivial permutation moves every element of that field rather than leaving any one of them in place.
A bridge between polynomials and permutations
The paper's algebraic framework uses a one-to-one correspondence between bivariate local permutation polynomials and q-tuples of permutations. That correspondence lets the authors study the polynomial constructions through associated permutation groups, linking each side of the problem to the other.
The companion construction is also explicit. The paper defines h and states that it yields a bivariate local permutation polynomial g that is a companion of f. The result supplies a paired polynomial within the same construction framework.
Equivalence is defined through the associated subgroups and conjugacy used in the paper. Its proposition on the new family states that it is not equivalent to any known family considered in the manuscript. The proof draws the distinction by contrasting the new non-Abelian subgroup in odd characteristic with Abelian or even-characteristic cases.
The paper counts the forms exactly
The main enumeration result is an exact closed-form count for the new family. Its proof divides the relevant conjugation conditions into disjoint sets, assigns p cubed elements to each nonempty set, and uses those pieces to calculate the normalizer needed for the count. The supplied extraction describes the resulting formula with factorials and powers of p. The paper presents the result as an exact enumeration rather than an estimate.
The paper also gives exact displayed counts for the three specified earlier families introduced in Lemmas 2.5, 2.6 and 2.7. The supplied extraction shows formulas involving factorials, powers and phi notation. It also leaves some superscript and fraction placement unclear, so the precise displays for those comparison families should be checked against the typeset source.
Separate propositions give the number of equivalent polynomials in each case. For the new family and for the families from Lemmas 2.5, 2.6 and 2.7, the paper reports the factorial of q equivalent permutation group polynomials. The count is attached respectively to the representative polynomials named f, f1, f2 and f3 in those results.
A result with a defined scope
The conclusions stay within the forms explicitly treated. The preprint reports a new odd-characteristic family, its explicit companion, inequivalence to the comparison families it considers, and exact enumeration for the specified constructions. No empirical sample or dataset is reported, so the result is an algebraic counting result rather than an observation about people, animals or laboratory systems.
The front matter identifies the manuscript as arXiv:2608.28118v1, dated 28 Aug 2026. The manuscript is a preprint.
Paper data and sources
Original title: Enumeration of certain permutation group polynomials
Authors: Sartaj Ul Hasan, Ramandeep Kaur, Hridesh Kumar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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