Two bijections for sets of words with forbidden factors

In a recent paper by Kitaev and Remmel, several formulas for the number of words of length n avoiding some generalized patterns were established. Each time the obtained function of n had been found in Sloane’s Encyclopedia as the number of some other objects, but the bijections between the two sets did not follow from the proof. Kitaev and Remmel stated four open problems on finding respective bijections. Here we solve two of them, concerning sequences A007070 and A048739.