A Combinatorial Interpretation of j/n {kn}\choose{n+j}
arXiv:math/0604471
Abstract
The identity j/n {kn}\choose{n+j} =(k-1) {kn-1}\choose{n+j-1}- {kn-1}\choose{n+j} shows that j/n {kn}\choose{n+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.
Latex, 7 pages, 5 figures