= Sparse backward links near full chain extension
{title2=$\Omega\simeq N^q/q!,\quad q=N_L$}
The exact count is $\Omega=\binom Nq=N^q\prod_{j=0}^{q-1}(1-j/N)/q!$. For fixed $q$ and large $N$, this is asymptotic to $N^q/q!$. Uniform relative accuracy requires $q^2/N\ll1$, since the logarithm of the product starts at $-q(q-1)/(2N)$. The weaker assumption $q/N\ll1$ gives the leading sparse-link entropy but need not give a small relative multiplicity error. In the continuum range $1\ll q\ll N$, the <entropic force> behaves as $(k_BT/(2a))\ln(N/q)$. The fixed-tension mean extension approaches $Na$ only as the tension diverges; at the discrete endpoint $q=0$ the exact entropy is zero and a continuous fixed-extension derivative is not defined.
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