Canon permutations and generalized descents of standard Young tableaux

Canon permutations are permutations of the multiset having k copies of each integer between 1 and n, with the property that the subsequences obtained by taking the jth copy of each entry, for each fixed j, are all the same. For k=2, canon permutations are sometimes called nonnesting permutations, and it is known that the polynomial that enumerates them by the number of descents factors as a product of an Eulerian polynomial and a Narayana polynomial. We extend this result to arbitrary k, and we relate the problem to the enumeration of standard Young tableaux of rectangular shape with respect to generalized descent statistics. Our proof is bijective, and it also settles a conjecture of Sulanke about the distribution of certain lattice path statistics.