A Combinatorial Interpretation of j/n knn+j

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The identity j/n knn+j =(k-1) kn-1n+j-1- kn-1n+j shows that j/n knn+j is always an integer. Here we give a combinatorial interpretation of this integer in terms of lattice paths, using a uniformly distributed statistic. In particular, the case j=1,k=2 gives yet another manifestation of the Catalan numbers.