3 problems

Let $I_{1}(n)$ denote the number of involutions in the symmetric group on $n$ symbols, let $p$ be an inefficient prime, and let $\nu_p(I_{1}(n))$ denote the $p$-adic valuation of…

Let $I_{1}(n)$ denote the number of involutions in the symmetric group on $n$ symbols, let $p$ be an odd inefficient prime, and let $\nu_p(I_{1}(n))$ be the $p$-adic valuation of…

2-adic shift conjecture. There is a 2-adic integer $\rho$ satisfying