It is shown that a formula that was independently obtained earlier for the number of cyclically irreducible words of length
n in a symmetric alphabet of a finitely generated free group of rank k and the Whitney formula for a chromatic polynomial
of a simple nonself-intersecting cycle of length n with a variable λ are mutually deducible from one another when λ = 2k.
The necessary bijections differ for even and odd values of n.
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