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A Smirnov word is a word over the positive integers in which adjacent lettersmust be different. A symmetric function enumerating these words by descentnumber arose in the work of Shareshian and the second named author on$q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville,and Vaughan for enumerating Smirnov words is proved. A symmetric functionenumerating a circular version of these words by cyclic descent number arose inthe work of the first named author on chromatic quasisymmetric functions ofdirected graphs, where a $t$-analog of a formula of Stanley for enumeratingcircular Smirnov words is proved.  In this paper we obtain new $t$-analogs of the Carlitz-Scoville-Vaughanformula and the Stanley formula in which the roles of descent number and cyclicdescent number are switched. These formulas show that the Smirnov wordenumerators are polynomials in $t$ whose coefficients are e-positive symmetricfunctions. We also obtain expansions in the power sum basis and the fundamentalquasisymmetric function basis, complementing earlier results of Shareshian andthe authors.  Our work relies on studying refinements of the Smirnov word enumerators thatcount certain restricted classes of Smirnov words by descent number.Applications to variations of $q$-Eulerian polynomials and to the chromaticquasisymmetric functions introduced by Shareshian and the second named authorare also presented.

On enumerators of Smirnov words by descents and cyclic descents