There are $n$ tennis players on a leader board, from rank $1$ (highest) to rank $n$ (lowest).

Every day, a match is held between a pair of players with adjacent ranks. When the higher rank player wins, nothing happens; otherwise, their ranks are exchanged, and we call that match an overtake by the winning player.

After $k$ days, the players find that all of them are back to their initial ranks. They then count the number of overtakes by each player.

Here is an example with $3$ players, named $A,B,C$ from highest to lowest initial rank.

Match	Winner	Loser	Rank $1,2,3$ after match	Overtake counts
				$A$	$B$	$C$
$1\ast$	$C$	$B$	$A,C,B$	$0$	$0$	$1$
$2$	$C$	$B$	$A,C,B$	$0$	$0$	$1$
$3\ast$	$C$	$A$	$C,A,B$	$0$	$0$	$2$
$4$	$A$	$B$	$C,A,B$	$0$	$0$	$2$
$5\ast$	$A$	$C$	$A,C,B$	$1$	$0$	$2$
$6\ast$	$B$	$C$	$A,B,C$	$1$	$1$	$2$

The matches marked with $\ast$ are overtakes.
After $6$ days, all players are back to initial ranks with overtake counts $1,1,2$.

Let $F(n,k)$ be the number of possible $n$-tuples of overtake counts after $k$ days, assuming that all players are back to initial ranks.
You are given $F(3,4)=8$ and $F(12,34)=2457178250$.

Find $F(123,4567891)mod1234567891$.