<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def count_matrix(dtraj, lag, sliding=True, sparse_return=True, nstates=None):
 r"""Generate a count matrix from given microstate trajectory. Parameters dtraj : array_like or list of array_like Discretized trajectory or list of discretized trajectories lag : int Lagtime in trajectory steps sliding : bool, optional If true the sliding window approach is used for transition counting. sparse_return : bool (optional) Whether to return a dense or a sparse matrix. nstates : int, optional Enforce a count-matrix with shape=(nstates, nstates) Returns ------- C : scipy.sparse.coo_matrix The count matrix at given lag in coordinate list format. Notes ----- Transition counts can be obtained from microstate trajectory using two methods. Couning at lag and slidingwindow counting. **Lag** This approach will skip all points in the trajectory that are seperated form the last point by less than the given lagtime :math:`\tau`. Transition counts :math:`c_{ij}(\tau)` are generated according to .. math:: c_{ij}(\tau) = \sum_{k=0}^{\left \lfloor \frac{N}{\tau} \right \rfloor -2} \chi_{i}(X_{k\tau})\chi_{j}(X_{(k+1)\tau}). :math:`\chi_{i}(x)` is the indicator function of :math:`i`, i.e :math:`\chi_{i}(x)=1` for :math:`x=i` and :math:`\chi_{i}(x)=0` for :math:`x \neq i`. **Sliding** The sliding approach slides along the trajectory and counts all transitions sperated by the lagtime :math:`\tau`. Transition counts :math:`c_{ij}(\tau)` are generated according to .. math:: c_{ij}(\tau)=\sum_{k=0}^{N-\tau-1} \chi_{i}(X_{k}) \chi_{j}(X_{k+\tau}). References .. [1] Prinz, J H, H Wu, M Sarich, B Keller, M Senne, M Held, J D Chodera, C Schuette and F Noe. 2011. Markov models of molecular kinetics: Generation and validation. J Chem Phys 134: 174105 Examples -------- Use the sliding approach first The generated matrix is a sparse matrix in CSR-format. For convenient printing we convert it to a dense ndarray. array([[ 1., 2.], [ 1., 1.]]) Let us compare to the count-matrix we obtain using the lag approach array([[ 0., 1.], [ 1., 1.]]) """

    # convert dtraj input, if it contains out of nested python lists to
    # a list of int ndarrays.
    dtraj = _ensure_dtraj_list(dtraj)
    return sparse.count_matrix.count_matrix_coo2_mult(dtraj, lag, sliding=sliding,
                                                      sparse=sparse_return, nstates=nstates)