Canonical partition function for DNA strands with open and closed links

[zexxa]

Question
Form the canoncial partition using the following conditions:

• 2 N-particles long strands can join each other at the i-th particle to form a double helix chain.
• Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open"
• An "open" link gives the strand $\epsilon$ amount of energy where $\epsilon > 0$
• A "closed" link gives the strand no energy
  
• For $m < N$, the strand must be "open" for $i \leq m$ and "closed" for $m < i \leq N$
• Note that $m \neq N$
• Each particle are independent of each other and they weakly interact

Attempt

I treated the energy states at $i /leq m$ and $i > m$ as a simple one energy state i.e.
$E = \sum_i ^N n_i \epsilon , {n_i} = \begin{cases} 1, & \text{if i \leq m}.\\ 0, & \text{otherwise} \end{cases}$
Therefore,
$Z(\beta , V , N) = \{ exp[ - \beta \epsilon ] \} ^m \{ exp[ 0 ]\} ^{N-m} = exp[ - \beta \epsilon ] ^m$

Does this make sense?

 

[DrClaude]

Does this make sense?

No. What is the base equation for the partition function?

 

[zexxa]

The base equation?
$Z ( \beta, V, N) = \sum_\alpha exp[ -\beta E ] = \sum_\alpha exp [ -\beta \sum_i^N n_i \epsilon ] = \sum_\alpha exp [ -\beta m \epsilon] = \{exp[ -\beta \epsilon ]\}^m = exp [-\beta \epsilon] ^m$

The base equation is the first 2 equations, and the rest are the manipulations I did to get to my answer from before.

 

[DrClaude]

$Z ( \beta, V, N) = \sum_\alpha exp[ -\beta E ] = \sum_\alpha exp [ -\beta \sum_i^N n_i \epsilon ] = \sum_\alpha exp [ -\beta m \epsilon] = \{exp[ -\beta \epsilon ]\}^m = exp [-\beta \epsilon] ^m$

I don't understand how you got from the 4th to the 5th terms in there. Also, shouldn't there be a link between $\alpha$ and $m$?