Let $T(n, k)$ (A008292) be the number of permutations of length $n+1$ of distinct elements $p_1,p_2,\cdots, p_{n+1}$, where the elements belong to the set $\left\lbrace1, 2,\cdots, n+1\right\rbrace$, $p_1=1$, and for which $$(k-1)(n+1) < \sum\limits_{j=1}^{n} (p_{j+1}-p_j)\bmod (n+1) < k(n+1)$$ when $1 < k < n$ and $T(n,1)=T(n,n)=1$.
How can this be proved?
By the way, $$T(n,k)=\left\langle\! n\atop k-1 \right\rangle\!$$
