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First, a quick note on terminology: the standard term for a "reverse order pair" is an inversion. Knowing this makes it easier to search for the answer: The generating function for the number of permu …

We have a Markov process where the state after $i$ steps is given by the size of the codomain of $g_i$. If at time $i$ we are in a state with $j$ surviving values, we can ignore the other values and c …

$$\begin{eqnarray*} \textrm{LHS} &=& \exp\left(\frac{a^2+b^2}2\right) \left(\frac{\partial^2}{\partial a^2} + \frac{\partial^2}{\partial b^2}\right)^m \exp\left(-\frac{a^2+b^2}2\right) \\ &=& \exp\lef …

We can compute the number of Dynkin systems for small $n$ using an almost-brute-force method. For efficient computation we represent a set ${a_i}$ as a binary number $\sum_i 2^{a_i}$; then $\Omega = \ …

$\det \begin{pmatrix} 1 \end{pmatrix} = 1$ works for any $p$. $\det \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} = -1$ similarly. For $n=3$ we require $p \ge 5$. By exhaustion there's no solution for …