Transposition changes the cycle count by one ($\ell (\tau \sigma )=\ell (\sigma )\pm 1$)

Left multiplication of a permutation by a transposition $(ab)$ swaps the two target labels $a,b$. If they occur in different cycles, $(a{x}_1\dots x_p)(b{y}_1\dots y_q)$ becomes the single cycle $(a{x}_1\dots x_{p}b{y}_1\dots y_q)$. If they occur in the same cycle, that formula reverses and splits the cycle in two. All other cycles are unchanged, so the total number of cycles, including fixed points, decreases or increases by exactly one. Empty intermediate lists yield one-cycles and require no exceptional case.