Trying to understand a proof of Laplace expansion I am trying to understand the ProofWiki proof of Laplace expansion for determinants.
I understand the first equation
$$
D = \sum_{\sigma} {\rm sgn} (\rho) {\rm sgn} (\sigma) \prod_{j=1}^n a_{\rho(j), \sigma(j)} 
$$
$D$ is the determinant of $n$ by $n$ matrix $(a_{i,j})$ and $\rho$ is any permutation.
but I don't understand the following step which states
$$
\small\sum_{\sigma}(−1)^{\tiny\displaystyle\sum^k_{i=1}(r_i+s_i)} {\rm sgn} (ρ(r_1,…,r_k)){\rm sgn} (σ(s_1,…,s_k)){\rm sgn} (ρ(r_{k+1},…,r_n)){\rm sgn} (σ(s_{k+1},…,s_n)) \prod_{j=1}^n a_{\rho(j), \sigma(j)} 
$$
Could someone kindly explain how to deduce this step? thank you
 A: The wikiproof page in question uses a rather cryptic notation. The main point is the following:
A partition of $\{1,...,n\}=H\sqcup H'$ into two subsets of (fixed) cardinalities $k$ and $n-k$, respectively, may be specified in a unique way through a permutation
$r(1,...,n)=(r_1,...,r_k, r_{k+1},...r_n)$ where the ordered lists
$r_1<\cdots < r_k$ and  $r_{k+1} < \cdots <r_n$ run through
the elements of the two subsets. Let $S_k(H) \times S_{n-k}(H')$ denote the set of permutations $\mu$ that shuffles elements within $H$ and within $H'$ but do not mix the two sets. Then  every permutation $\rho\in S_n$ may be written in a unique way as $\rho=\mu\circ r$ for some $r$ and $\mu$  of the above-mentioned form.
An example with $n=5,k=2$ : Given the permutation
$(1, 2, 3, 4, 5)\stackrel{\rho}\longrightarrow 
 (5, 2; 4, 1, 3)$ (the semi-colon is just to underline the partitions) there is a unique splitting
(with $H=\{2,5\}$, $H'=\{1,3,4\}$):
$$(1, 2, 3, 4, 5) 
\stackrel{r}{\longrightarrow}(2, 5; 1, 3, 4)
 \stackrel{\mu}\longrightarrow 
 (5, 2; 4, 1, 3).
 $$
In the equation you mention $\rho$ is any permutation but  e.g. $\rho(r_1,...,r_k)$ is what I would prefer to call $\mu(r_1,...,r_k)$, since $\mu$ restricted to $\{r_1,...,r_k\}$ is indeed just a permutation of that set. Anyway, as  $\mu$ is just the product of two permutations the signature of $\mu$ is the product of signatures (in the article writing $\rho$ instead of $\mu$), $$ {\rm sgn}(\mu(r_1,...,r_k)) \times {\rm sgn}(\mu (r_{k+1},...,r_n)).$$
For the signature of $r$ in my example you need $r_2-2= 5-2=3$ neighboring binary swaps to move five from position 2 to position 5 and then you need $r_1-1=1$ binary swap to move $2$ to position 2.
Note that automatically this puts the remaining $n-k=3$ elements in the correct order!
In general, the number of binary swaps is given by
$$\sum_{i=1}^k (r_i-i)=\sum_{i=1}^k r_i - k(k+1)/2.$$ Thus,
$${\rm sgn}(r)=(-1)^{\sum_{i=1} r_i} \times (-1)^{k(k+1)/2}.$$
Now use the same argument for the permutation of the
second indices, $\sigma\circ s$. The last factor appears twice, but as
$(-1)^{k(k+1)}=1$  it disappears and you obtain the formula you stated when you make the above interpretation of $\rho(r_1,...,r_k)$ etc. The sequel of the article
makes sense when using this interpretation. Hope this helps.
