3 problems

For $n\geq 1$, define $B(n,k)$ and $C(n,k)$ by … and … Here a simsun permutation is a permutation with the stated simsun property, and a big return is the stated statistic on an al…

Alternating Turán-type inequality. For all $n\geq 0$ and $j,k\geq 1$, one has

Asymptotic density conjecture. For every $1\leq\alpha\leq 6$, the limit $p_{\infty}^{(\alpha)}=\lim_{n\rightarrow\infty}p_n^{(\alpha)}$ exists and satisfies