All Questions

This is a continuation of my post Number of bounded Dyck paths with "negative length". Let $C_{n}^{(2k+1)}$ be the number of Dyck paths of semilength $n$ bounded by $2k+1.$ They satisfy a ...

Let ‎$‎n‎$ ‎be a‎ ‎positive ‎integers and ‎$‎T=T_{n,n}‎$ ‎be the ‎$‎n\times n‎$‎ table in the first quadrant composed of $n^2$ unit squares‎, ‎whose $(x,y)$-blank is locate in the $x^{th}$-column from ...

Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity. ...