How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well).


If only every row (or column) is distinct is needed, the answer is easy.


As suggested in comments, <https://oeis.org/A088310> provides the numbers without symmetric restriction. I do not see a proof or link for proof there.


What about symmetric case?



Counting matrices of special types
For generic (not necessarily symmetric) $m\times n$ matrices over a set of $k$ elements, the number of those with pairwise distinct columns and rows is
$$\sum\_{i=0}^m\sum\_{j=0}^n s(m,i)\cdot s(n,j)\cdot k^{i\cdot j},$$
where $s(,)$ are Stirling numbers of first kind with sign.


**UPDATE**. For symmetric $n\times n$ matrices over a set of $k$ elements, the number of those with pairwise distinct columns and rows is
$$\sum\_{i=0}^n s(n,i)\cdot k^{i(i+1)/2}.$$


For k=2, numerical values for $n=1,2,\dots$ are
$$2, 6, 44, 716, 24416, 1680224, 229468288, \dots$$
and now form the sequence [A259763](http://oeis.org/A259763) in the OEIS. Just in case, I have verified these values for $n\leq 5$ by a direct enumeration of matrices.


