Alternative representation for $B_{n,k}(1!,\dots,(n-k+1)!)$

Question

Let $B_{n,k}$ be the Exponential Bell Polynomial and $\hat {B}_{n,k}$ the Ordinary Bell Polynomial. One has the identity: $$ B_{n,k}(0!,1!,\dots,(n-k)!) = |s(n,k)|, $$ with $s(n,k)$ the Stirling number of the first type. I am however interested in $B_{n,k}(1!,2!,\dots,(n-k+1)!)$, note that this satisfies: $$ B_{n,k}(1!,2!,\dots,(n-k+1)!) = \hat B_{n,k}(1,\dots,1), $$ this seems like a very simple formula but I do not find any references linking this quantity to Sterling numbers, Bell numbers or any other type of numbers which I find odd.

Answer 1

They are the Lah numbers(number of partitions of $[n]$ into $[k]$ linearly ordered sets): $$\frac{n!}{k!}\binom{n-1}{k-1}.$$ Notice that the difference in between the signed Stirling and this ones is that one is a cyclic order and the other is a linear order. They differ by a factor of $k.$ Which is the shifting you want.