Jang Soo Kim (김장수), Longest elements in a semigroup of functions and Slater indices

The group $S_n$ of permutations on $[n]=\{1,2,\dots ,n\}$ is generated by simple transpositions $s_i=(i,i+1)$. The length $\ell (\pi )$ of a permutation $\pi$ is defined to be the minimum number of generators whose product is $\pi$. It is well-known that the longest element in $S_n$ has length $n(n-1)/2$. Let $F_n$ be the semigroup of functions $f:[n]\to [n]$, which are generated by the simple transpositions $s_i$ and the function $t:[n]\to [n]$ given by $t(1)=t(2)=1$ and $t(i)=i$ for $i\ge 3$. The length $\ell (f)$ of a function $f\in F_n$ is defined to be the minimum number of these generators whose product is $f$. In this talk, we study the length of longest elements in $F_n$. We also find a connection with the Slater index of a tournament of the
complete graph $K_n$. This is joint work with Yasuhide Numata.