### Operations on partially ordered sets and rational identities of type A

*Adrien Boussicault*

#### Abstract

We consider the family of rational functions ψ

_{w}= ∏( x_{wi}- x_{wi+1})^{-1}indexed by words with no repetition. We study the combinatorics of the sums Ψ_{P}of the functions ψ_{w}when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction Ψ_{P}. We prove that the denominator of Ψ_{P}has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of Ψ_{P}can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.Full Text: PDF PostScript