Permutation Descent Counts

Given a positive integer, $N$, a permutation of order $N$ is a one-to-one (and thus onto) function from the set of integers from $1$ to $N$ to itself. If $p$ is such a function, we represent the function by a list of its values:

\[ [ \, p(1) \, p(2) \, \ldots \, p(N) \, ] \]For example, $[5 \, 6 \, 2 \, 4 \, 7 \, 1 \, 3]$ represents the function from ${ 1 \ldots 7 }$ to itself which takes $1$ to $5$, $2$ to $6$, ... , $7$ to $3$.

For any permutation $p$, a descent of $p$ is an integer $k$ for which $p(k) > p(k+1)$. For example, the permutation $[5 \, 6 \, 2 \, 4 \, 7 \, 1 \, 3]$ has a descent at $2$ ($6 > 2$) and $5$ ($7 > 1$).

For permutation $p$, $\text {des}(p)$ is the number of descents in $p$. For example, $\text {des}([5 \, 6 \, 2 \, 4 \, 7 \, 1 \, 3]) = 2$. The identity permutation is the only permutation with $\text {des}(p) = 0$. The reversing permutation with $p(k) = N + 1 - k$ is the only permutation with $\text {des}(p) = N - 1$.

The permutation descent count (PDC) for given order $N$ and value $v$ is the number of permutations $p$ of order $N$ with $\text {des}(p) = v$. For example:

PDC(3, 0) = 1 { [ 1 2 3 ] } PDC(3, 1) = 4 { [ 1 3 2 ], [ 2 1 3 ], [ 2 3 1 ], 3 1 2 ] } PDC(3, 2) = 1 { [ 3 2 1 ] }

Write a program to compute the PDC for inputs $N$ and $v$. To avoid having to deal with very large numbers, your answer (and your intermediate calculations) will be computed modulo $1001113$.

The first line of input contains a single integer $P$, ($1 \le P \le 1\, 000$), which is the number of data sets that follow. Each data set should be processed identically and independently.

Each data set consists of a single line of input. It contains the data set number, $K$, followed by the integer order, $N$ ($2 \le N \le 100$), followed by an integer value, $v$ ($0 \le v \le N-1$).

For each data set there is a single line of output. The single output line consists of the data set number, $K$, followed by a single space followed by the PDC of $N$ and $v$ as a decimal integer number modulo $1001113$.

Sample Input 1 | Sample Output 1 |
---|---|

4 1 3 1 2 5 2 3 8 3 4 99 50 |
1 4 2 66 3 15619 4 325091 |