Operations on posets and rational identities of type A

(xi − xj) . The patterns appear in geometry, characterizing some families of Schubert varieties. Schubert varieties are indexed by permutations, and the varieties which are non singular are those whose indexing permutation does not contain the pattern 2143 nor the pattern 1324. In [2], Cortez has described geometrical properties of Schubert varieties for permutations avoiding the patterns 1324 and 2143. This was further clarified by Woo and Yong in [6], and Butler and Bousquet-Mélou in [1]. They use the fact that Hasse diagram naturally associated to a permutation avoiding 1324 and 2143 is acyclic. Surprisingly, the same patterns will occur in the study of our rational functions. In fact our work is closely connected to a study of Greene [3] on rational identity related to Murnaghan-Nakayama formula for Sn (type A). Greene gives in [3] a closed expression for the sum ΨP of the ψw where the indices are taken in the linear extensions L(P ) of a planar poset P ,