Assume Nakayama algebras are connected and given by quiver and relations. Note that such Nakayama algebras have global dimension at most $2n-2$ in case it is finite and the algebra has $n$ simples.
For $n$ fixed and $1 \leq k \leq 2n-2$ let $a_{n,k}$ be the minimal vector space dimension of a Nakayama algebra with $n$ simples and global dimension $k$.
Question 1: Do we have $a_{n,i} \geq a_{n,i+1}$ for $i=1,...,n-1$ and $a_{n,i} \leq a_{n,i+1}$ for $i=n-1,...,2n-2$?
(What is the name for such sequences by the way? Reverse unimodal?)
Here $a_{n,k}$ for $n=8$: [ 36, 18, 17, 16, 16, 16, 15, 17, 20, 23, 29, 35, 57, 71 ]
For $n$ fixed and $1 \leq k \leq 2n-2$ let $b_{n,k}$ be the maximal vector space dimension of a Nakayama algebra with $n$ simples and global dimension $k$.
Question 2: Do we have $b_{n,i} < b_{n,i+1}$ for $i$ odd and $b_{n,i}>b_{n,i+1}$ for $i$ even?
Here $b_{n,k}$ for $n=8$:
[ 36, 92, 77, 91, 71, 85, 66, 80, 62, 76, 59, 73, 57, 71 ]
Both questions have a positive answer for $n \leq 9$.
