TITLE: Multi Index Dirac delta function
QUESTION [1 upvotes]: If we assume the following result:
$$\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k, \rho}_{\beta_1,\beta_2,\cdots , \beta_k, \rho} = (n-k)\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k}_{\beta_1,\beta_2,\cdots , \beta_k}$$
Apparently I'm supposed to show
$$\delta^{\rho_1,\rho_2,\cdots , \rho_k}_{\rho_1,\rho_2,\cdots , \rho_k} = (n-k+1)\cdots (n)$$
But I keep getting $=(n-k+1)\cdots (n-1)$.
As a concrete example, consider
$$\delta^{\rho_1,\rho_2}_{\rho_1, \rho_2}=(n-1)\delta_{\rho_1}^{\rho_1}=(n-1)\neq n$$

REPLY [0 votes]: As per the recommendation of a comment, I'll answer this.  
In general $\delta^j_i$ is $1$ if $j=i$ and $0$ otherwise.  
In general, if we write the same symbol we sum over that symbol.
If we write the same symbol, the two symbols are obviously equal but if we don't write the same symbol they may or may not be equal to each other.
With these ideas in mind, we easily see that $\delta_i^i= n$
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