Bijective Link Between Chapoton’s New Intervals and Bipartite Planar Maps

In 2006, Chapoton defined a class of Tamari intervals called « new intervals » in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called « degree tree ». Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.