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The 2-ordinal polytopes associated with the symmetric groups are the permutohedra. The number of codimension $k$ faces of the $n$th permutohedron is given in sequence A019538.

The second diagonal has some nice properties. For example, Alexander Povolotsky observed that these numbers $P_{i}$ arise as the right hand coefficients for the following sequence of expressions, indexed by $i$.

$n(n+1)[n + (n+1)] = 6(1 + 4 + 9 + \cdots + n^2 )$
$n(n+1)(n+2)[n + (n+1) + (n+2)] = 36(1 + (1+4) + (1+4+9) + \cdots + (1 + 4 + \cdots + n^2 ))$

This brings to mind the Leech sequence

$1 + 2^2 + 3^2 + \cdots + 24^{2} = 70^{2}$

for $n=24$, for which the first element of the list is expressed

$\frac{1}{3}(\sum_{i=1}^{n} i ) (2n + 1) = n^{2}$

If the squares of integers up to $n > 24$ cannot be summed to a square, it follows that the left hand side can never be a square.