Product of cycles and a transposition
Dependencies:
Let $\sigma_1 = (a_1, a_2, \ldots, a_n)$ and $\sigma_2 = (b_1, b_2, \ldots, b_m)$ be two disjoint cycles. Let $\tau$ be a transposition.
Case 1: $\tau$ is disjoint to both cycles
No further simplification of products is possible.
Case 2: 1 element of $\tau$ intersects $\sigma_1$, the other element intersects neither $\sigma_1$ nor $\sigma_2$
Let $\tau = (a_i, c)$.
\[ \sigma_1 \tau = (a_1, a_2, \ldots, a_n) (a_i, c) = (a_1, a_2, \ldots, a_i, c, a_{i+1}, \ldots, a_n) \]
\[ \tau \sigma_1 = (a_i, c) (a_1, a_2, \ldots, a_n) = (a_1, a_2, \ldots, a_{i-1}, c, a_i, \ldots, a_n) \]
Each of $\sigma_1\tau$ and $\tau\sigma_1$ is a cycle of length $n+1$.
Case 3: Both elements of $\tau$ intersect $\sigma_1$
Let $\tau = (a_i, a_j)$, where $i < j$.
\[ \sigma_1 \tau = (a_1, a_2, \ldots, a_n) (a_i, a_j) = (a_1, a_2, \ldots, a_i, a_{j+1}, \ldots, a_n) (a_{i+1}, a_{i+2}, \ldots, a_j) \]
\[ \tau \sigma_1 = (a_i, a_j) (a_1, a_2, \ldots, a_n) = (a_1, a_2, \ldots, a_{i-1}, a_{j}, \ldots, a_n) (a_i, a_{i+1}, \ldots, a_{j-1}) \]
For both $\sigma_1\tau$ and $\tau\sigma_1$, we get two disjoint cycles of length $j-i$ and $n-(j-i)$.
When $j = i+1$, we get a single cycle of length $n-1$.
Case 4: $\tau$ intersects both cycles
Let $\tau = (a_i, b_j)$.
\[ \sigma_1\sigma_2\tau = (a_1, a_2, \ldots, a_n) (b_1, b_2, \ldots, b_m) (a_i, b_j) = (a_1, \ldots, a_i, b_{j+1}, \ldots, b_m, b_1, \ldots, b_j, a_{i+1}, \ldots, a_n) \]
\[ \sigma_1\tau\sigma_2 = (a_1, a_2, \ldots, a_n) (a_i, b_j) (b_1, b_2, \ldots, b_m) = (a_1, \ldots, a_i, b_j, \ldots, b_m, b_1, \ldots, b_{j-1}, a_{i+1}, \ldots, a_n) \]
\[ \tau\sigma_1\sigma_2 = (a_i, b_j) (a_1, a_2, \ldots, a_n) (b_1, b_2, \ldots, b_m) = (a_1, \ldots, a_{i-1}, b_j, \ldots, b_m, b_1, \ldots, b_{j-1}, a_i, \ldots, a_n) \]
For all three of these products, we get a single cycle of length $m+n$.
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- Number of transitive dependencies: 5
Transitive dependencies:
- /sets-and-relations/relation-composition-is-associative
- /sets-and-relations/composition-of-bijections-is-a-bijection
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