Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $0/1$ entries.

We have:

$$\mathcal R_2[1]=\mathcal R[1]+\mathcal R[1]=\{0,1\}+\{0,1\}=\{0,1,2\}\subseteq\mathcal R[2]$$

$$\mathcal R_2[2]=\mathcal R[2]+\mathcal R[2]=\{0,1,2\}+\{0,1,2\}=\{0,1,2,3,4\}\subseteq\mathcal R[3]$$

$$\mathcal R_2[3]=\mathcal R[3]+\mathcal R[3]=\{0,1,2,3,4,6\}+\{0,1,2,3,4,6\}=\{0,1,2,3,4,5,6,7,8,9,10,12\}\subseteq\mathcal R[4]$$

However $\mathcal R_2[n]\not\subseteq\mathcal R[n+1]$?

Counter example: $27,35\in\mathcal R_2[4]\cap \mathcal R[5]^c$. Also check OEIS A089477.

Denote $R_2'[n]=\{i\in\mathcal R_2[n]:i\in\mathcal R[n+1]\}$.

Is there an $\epsilon>0$ such that $(2-\epsilon)|R_2'[n]|>|R_2[n]|$ holds for all $n$ bigger than some $n_0\in\Bbb N$?

If there is no $\epsilon>0$ then what is the slowest growing function $f(n)$ such that $f(n)|R_2'[n]|>|R_2[n]|$ holds for all $n$ bigger than some $n_0\in\Bbb N$?

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