A Robinson–Schensted Algorithm for a Class of Partial Orders

LetPbe a finite partial order which does not contain an induced subposet isomorphic with 3+1, and letGbe the incomparability graph ofP. Gasharov has shown that the chromatic symmetric functionXGhas nonnegative coefficients when expanded in terms of Schur functions; his proof uses the dual Jacobi–Trudi identity and a sign-reversing involution to interpret these coefficients as numbers ofP-tableau. This suggests the possibility of a direct bijective proof of this result, generalizing the Robinson–Schensted correspondence. We provide such an algorithm here under the additional hypothesis thatPdoes not contain an induced subposet isomorphic with {x>a<b<c>y}.