Number of cyclically irreducible words in the alphabet of a free group of finite rank

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. To the memory of William T. Tutte (05.14.1917–05.02.2002) __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 39–48, July–August 2007.