Combinatorial argument for irreducible polynomials over GF[2]

If we have a sequence of pairs (1,a_1), (2,a_2), .... , (n, a_n) such that the sequence a_1, ... , a_n first increases and decreases, and the n pairs can be rearrange into: (b_1, b_2), (b_2, b_3), .... , (b_{n-1}, b_n), (b_n, b_1).

I have this conjecture that the number of these sequences of pairs are equal to the number of irreducible polynomials over GF[2]. I kinda see that the proof goes in the direction of considering the fact that the roots of any irreducible polynomial form a single orbit under the action of the multiplicative group. How might I approach the proof?