There is a typo. There should be 0=k_0<k_1<\dots < k_t = N, where N is the number of shares in the PPLNS window such that the m-slice contains the shares s_{k_{m-1}+1}, \dots , s_{k_{m}}. Recall that, by Remark 5 in Section 3.2, if S is a slice, we have that \sum_{s \in S} score_f(s) =1. Hence
\sum_{j=k_0+1}^{k_1} score_f(s_j) + ... + \sum_{j=k_{t-1}+1}^{k_t} score_f(s_j) = 1+\dots +1 =t