Splits

For a permutation $p = p\[0] p\[1] p\[2] \dots p\[n − 1]$ of the numbers $1, 2, 3,\dots,n$ we define a split as a permutation $q$ which can be obtained by the following process:

1. Select two sets of numbers $A = \\{ i\_1 ,i\_2 , \dots , i\_k \\}$ and $B = \\{ j\_1 , j\_2 ,\dots , j\_l \\}$ such that $A ∩ B = ∅$, $A ∪ B = \\{ 0, 1, 2, \dots ,n − 1 \\}$, $i\_1 < i\_2 < \dots < i\_k$ and $j\_1 < j\_2 < \dots < j\_l$.
2. The permutation $q$ will be $q = p\[i\_1 ]p\[i\_2 ]\dots p\[i\_k ]p\[j\_1 ]p\[j\_2 ]\dots p\[j\_l ]$

Moreover, we define $S(p)$ to be the set of all splits of a permutation $p$.

You are given a number $n$ and a set $T$ of $m$ permutations of length $n$. Count how many permutations $p$ of length $n$ exist such that $T ⊆ S(p)$. Since this number can be large, find it modulo $998\\, 244\\, 353$.

• $1 ≤ n ≤ 300$
• $1 ≤ m ≤ 300$