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The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by $$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$ Definition. Fix an integer $m\geq0$....

Question Let $g$ to be an element of permutation group $S_n$, and $\tau = (1,2,3,\cdots,n)$ is the circular permutation. $g$ and $\tau g$ have $n+1$ cycles in total(fixed point is also a cycle), ...

This is a follow-up question of my SO post I'll briefly mention it here. So given a $n$ cycle say $(1,2,\ldots,n)$, what are the monotonic 2 -tuples, of the form $(a,b)(c,d)$, monotonicity in on the ...