Cauchy Number of a Permutation

Question

Find the Cauchy number for the permutation $(1354)(267)$.

I did a google search on the definition of a Cauchy number, but nothing really came up. This is the best I could find, but it is rather obscure. Does anyone know the definition?

Answer 1

Found here

If the symmetric group $S_n$ is split into $n$ disjoint subgroups of orders $a_1, a_2...a_n$, the Cauchy number is $\sum _{i=1}^n (a_i-1)$

In your case $(1354)(267)$ means two subgroups of order $4!$ and $3!$, then following the definition the Cauchy number should be $(4!-1)+(3!-1)=28$

$$\left( \begin{array}{ccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 6 & 5 & 1 & 4 & 7 & 2 \\ \end{array} \right)$$

Hope it's not too wrong :)