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Let $(a_n)$ be the sequence of minimum values of the expression (depending on $\ell$):

\begin{equation*} a_n=\min \bigg\lbrace ((\ell+1)j-n)!\,(n-\ell j)!\quad \text{for}\; \\,\bigg\lceil\frac{n}{\ell+1}\bigg\rceil\le j\le \bigg\lfloor\frac{n}{\ell}\bigg\rfloor \bigg\rbrace % \min \big\lbrace (2j-n)!\,(n- j)!,\; \mathrm{ceil}\,\frac{n}{2}\le j\le n \big\rbrace, .\end{equation*}

My question is twofold:

Is there a closed formula for $(a_n)_{n\ge \ell}$ (for a fixed $\ell$), or a way to characterize its entries? Why?

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You asked this at math.stackexchange.com/questions/180374/… some 20 hours ago. It is usually best to ask on one site and wait a bit (a few days) before reasking on the other site. When you do ask again, add links between the two instances of the question. –  Mariano Suárez-Alvarez Aug 9 '12 at 16:42
    
You can write a program, with Maple, Magma or Matlab, and obtaining some directions for examining. Also, if you give these results here, I think it is helpful and the problem will be clearer. –  Shahrooz Aug 9 '12 at 17:35
    
This is for fixed l, (in my view) a fairly simple optimization, and is equivalent to the minimum of a set of two values. I would like more motivation, an idea of what you have tried, and either admission that it is homework or a real good reason why I should not let you solve it yourself. Gerhard "Willing To Give Another Hint" Paseman, 2012.08.09 –  Gerhard Paseman Aug 9 '12 at 18:16
    
I posted a similar question at StackExchange: math.stackexchange.com/questions/180374/… –  daniel birmajer Aug 10 '12 at 17:23
    
One thing that bothers me about the question is that it is ill defined. For example, a_(l+2) does not exist for most values of l, and this is true of some other indices. Further, a computer search using valid pairs of (l,n) should make it clear which values of j will produce the desired minimum, and such j can be easily described. Because of the nature of the description, I suspect no nice form exists for the general term. Gerhard "Ask Me About System Design" Paseman, 2012.08.10 –  Gerhard Paseman Aug 10 '12 at 18:15

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