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On the Carlitz rank of permutation polynomials. (English) Zbl 1232.11124
Summary: A well-known result of L. Carlitz [Proc. Am. Math. Soc. 4, 538 (1953; Zbl 0052.03704)], that any permutation polynomial $$p(x)$$ of a finite field $$\mathbb F_q$$ is a composition of linear polynomials and the monomial $$x(q-2)$$ implies that $$p(x)$$ can be represented by a polynomial $$P_n(x) = (...((a(0)x + a(1))(q-2) + a(2))(q-2)...+ a(n))(q-2) + a(n+1)$$. for some $$n\geq 0$$. The smallest integer $$n$$, such that $$P_n(x)$$ represents $$p(x)$$ is of interest since it is the least number of “inversions” $$x(q-2)$$, needed to obtain $$p(x)$$. We define the Carlitz rank of $$p(x)$$ as $$n$$, and focus here on the problem of evaluating it. We also obtain results on the enumeration of permutations of $$\mathbb F_q$$ with a fixed Carlitz rank.

##### MSC:
 11T06 Polynomials over finite fields
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##### References:
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