Billey, Sara C.Mitchell, Stephen C.2013-02-012013-02-012008http://hdl.handle.net/1773/20987Below is the code used to supplement the proofs in "Affine Partitions and Affine Grassmannians" by Sara Billey and Stephen Mitchell. The code includes algorithms for generating elements in Coxeter groups up to some length, the Coxeter matrices for Weyl groups and Affine Weyl groups, algorithms for quotients of Coxeter groups, affine partitions, colored partitions, rank generating functions, Bruhat order, weak order, generalized Young's lattice, etc. The code supplements the proofs in the paper by proving that the affine partitions in each exceptional type are equinumerous with the minimal length coset representatives for the affine Weyl group mod the Weyl group. The lisp code can be used to identify the generating function for affine partitions. The maple code takes in this generating function, simplifies it, and compares it with Bott's formula.We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type An, the affine colored partitions form another family of combinatorial objects in bijection with n + 1 core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young’s lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.Article