2

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that $$\mathrm{Pr}[\sigma_1\sigma_2\cdots\sigma_k=\sigma_k\sigma_{k-1}\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}?$$

  • Importantly, $2/n!$ is the probability of two random permutations being equal conditioned on them being the same sign.

Experimental results give the following table (after $10^6$ samples):

\begin{array}{r|rrrr} & n=5 & 6 & 7 & 8 \\ \hline k=1 & 1.000000 & 1.000000 & 1.000000 & 1.000000 \\ 2 & 0.058090 & 0.015159 & 0.002950 & 0.000492 \\ 3 & 0.058050 & 0.015375 & 0.002874 & 0.000552 \\ 4 & 0.018606 & 0.003231 & 0.000416 & 0.000058 \\ 5 & 0.018796 & 0.003131 & 0.000385 & 0.000057 \\ 6 & 0.016813 & 0.002763 & 0.000403 & 0.000053 \\ 7 & 0.016768 & 0.002768 & 0.000426 & 0.000046 \\ 8 & 0.016818 & 0.002711 & 0.000406 & 0.000053 \\ 9 & 0.016589 & 0.002736 & 0.000397 & 0.000050 \\ 10 & 0.016515 & 0.002792 & 0.000390 & 0.000061 \\ 11 & 0.016478 & 0.002685 & 0.000385 & 0.000054 \\ 12 & 0.016757 & 0.002834 & 0.000363 & 0.000056 \\ 13 & 0.016550 & 0.002835 & 0.000389 & 0.000049 \\ 14 & 0.016848 & 0.002850 & 0.000403 & 0.000049 \\ 15 & 0.016680 & 0.002803 & 0.000383 & 0.000050 \\ \hline 2/n! \simeq & 0.016667 & 0.002778 & 0.000397 & 0.000050 \end{array}

This looks like fairly strong evidence that the limit holds.

This question relates to two previous questions of mine:

where it is proved that $$\mathrm{Pr}[\sigma_1\sigma_2\cdots\sigma_k=\sigma_k\sigma_{k-1}\cdots\sigma_1]=\frac{\text{nr conjugacy classes of } S_n}{n!}=\frac{\text{nr partitions of } n}{n!}$$ when $k \in \{2,3\}$.

Community
  • 1
Rebecca J. Stones
  • 26,845
  • My first thought was to use the formula at the top of page 4 in this document. However it's incorrect (the LHS is $\le1$ while the RHS is $\ge1$). I went to the source material to better understand, and based on my understanding of (35) & (46), the correct formula is $\frac{1}{|G|}\sum_\rho(\dim\rho)^\kappa$, where $\kappa$ is the Euler characteristic of the cell complex $X(w)$ associated to the word $w=a_1\cdots a_n(a_{\sigma(1)}\cdots a_{\sigma(n)})^{-1}$. – anon Jul 02 '14 at 23:16
  • The author in the first link has the powers of the $\dim$s to be $k-1$ where $k$ is the minimal number of "block transpositions" in a factorization of $\sigma$. I don't know how to connect $\kappa$ and $k-1$ (I am not familiar with the relevant topology), and I don't know a good formula for $k$ in the case that $\sigma=(n\cdots 1)$ (in one-line notation), but that's the route I'd go on. – anon Jul 02 '14 at 23:17

0 Answers0