Operations on partially ordered sets and rational identities of type A
Adrien Boussicault
Abstract
We consider the family of rational functions ψw=
∏( xwi - xwi+1 )-1 indexed by
words with no repetition. We study the combinatorics of the sums
ΨP of the functions
ψw 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
ΨP. We prove that the denominator of
ΨP 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
ΨP can be reduced to the case of bipartite
posets. Finally, we compute the numerators associated to some special
bipartite graphs as Schubert polynomials.
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