Timeline for Minimum number of transpositions to make two multiset permutations equal

8 events

when toggle format	what		by	license	comment
Jan 30 at 9:16	history	edited	caduk	CC BY-SA 4.0	deleted 1 character in body
Jan 16 at 7:11	vote	accept	Fabius Wiesner
Jan 15 at 21:39	comment	added	Luc Guyot		... cannot be the product of less than $m(n - 1)$ transpositions.
Jan 15 at 21:36	comment	added	Luc Guyot		Many thanks for your input. I think I get the second part: there is a surjective forgetful map from $\text{Sym}(mn)$ (the labeled permutations) onto the permutations of the OP's multiset (those are label-less permutations), that is, the natural map $\text{Sym}(mn) \rightarrow \text{Sym}(mn) / \text{Sym}(m)^n$. Then you exhibit a special element $\sigma$ such that $\sigma \theta$ has exactly $m$ cycles for every $\theta \in \text{Sym}(m)^n$. This means that the reflection length of $\sigma \theta$ is $m(n - 1)$ for every $\theta$. Hence $\sigma$, as a permutation of the multiset ...
Jan 15 at 13:52	comment	added	caduk		@FabiusWiesner done, there was a slight mistake, we can have less than $m$ cycles (but not more)
Jan 15 at 13:51	history	edited	caduk	CC BY-SA 4.0	added 119 characters in body
Jan 15 at 10:45	comment	added	Fabius Wiesner		Could you give an example for the last statement?
Jan 15 at 9:00	history	answered	caduk	CC BY-SA 4.0