We have two conjectured generalizations of the question asked at
[a Hankel matrix of involution numbers](https://mathoverflow.net/questions/261600/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?



Further aspects of a Hankel matrix of involution numbers
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:


$$ \left[ \begin {array}{ccccc} 1&1&2&4&10\\ 1&2&4&10&
26\\ 2&4&10&26&76\\ 4&10&26&76&232
\\ 10&26&76&232&764\end {array} \right] =$$$$= \left[ \begin {array}{ccccc} 1&&&&\\ 1&1&&&
\\ 2&2&1&&\\ 4&6&3&1& 
\\ 10&16&12&4&1\end {array} \right] \left[ \begin {array}{ccccc} 1& & & & \\ &1& & & 
\\ & &2& & \\ & & &6& 
\\ & & & &24\end {array} \right] \left[ \begin {array}{ccccc} 1&1&2&4&10\\ &1&2&6&
16\\ & &1&3&12\\ & & &1&4
\\ & & & &1\end {array} \right] 
$$


