TITLE: Further aspects of a Hankel matrix of involution numbers
QUESTION [11 upvotes]: We have two conjectured generalizations of the question asked at
a Hankel matrix of involution numbers
by Tewodros Amdeberhan. Let $n!!=1!\,2!\cdots n!$.
Conjecture 1. Let $I_k$ denote the number of involutions in the
  symmetric group $\mathfrak{S}_k$. Then the Smith normal form of the
  matrix $[I_{i+j}]_{i,j=0}^n$ has diagonal entries $0!, 1!, \dots,
  n!$,
Conjecture 2. Let $J_k=k!\sum_{\lambda\vdash k}s_\lambda$, where
  $s_\lambda$ is a Schur function. Set $J_0=1$. When the symmetric
  function $\det[J_{i+j}]_{i,j=0}^n$ is expanded in terms of power
  sums, then every coefficient is an integer divisible by
  $n!!$. (Tewodros' question is equivalent to the coefficient of
  $p_1^{n(n+1)}$ being equal to $n!!$.)
Is there a nice formula or combinatorial interpretation of the
coefficients in Conjecture 2?

REPLY [3 votes]: On Conjecture 1: As remarked by Johann Cigler in the  linked question (and shown in the references linked there) the matrix $M:=\left[I_{i+j}\right]_{i\ge0\atop j\ge0}$ diagonalises as $M=U^TDU$ with $D:=\operatorname{diag(k!)}$, and $U$ an upper triangular integer coefficients matrix with unit diagonal elements, hence with integer coefficient inverse: so  this is also the SNF of $M$.
For example:
$$ \left[ \begin {array}{ccccc} 1&1&2&4&10\\  1&2&4&10&
26\\ 2&4&10&26&76\\  4&10&26&76&232
\\  10&26&76&232&764\end {array} \right] =$$$$= \left[ \begin {array}{ccccc} 1&&&&\\ 1&1&&&
\\  2&2&1&&\\  4&6&3&1& 
\\  10&16&12&4&1\end {array} \right] \left[ \begin {array}{ccccc} 1& & & & \\   &1& & & 
\\   & &2& & \\   & & &6& 
\\   & & & &24\end {array} \right] \left[ \begin {array}{ccccc} 1&1&2&4&10\\   &1&2&6&
16\\   & &1&3&12\\   & & &1&4
\\   & & & &1\end {array} \right] 
$$