k.vexillary permutations

The following permutations are minimal patterns which are not k-vexillary. These patterns supplement the paper "Permutation Patterns, Stanley Symmetric Functions and Generalized Specht Modules" by Sara Billey and Brendan Pawlowski. The patterns characterizing 2-vexillary, 3-vexillary and 4-vexillary permutations are known to be a complete set. We conjectured that the 2346 patterns in the list *4-vex* give a complete characterization in our paper. Michael Albert and David Eyers deployed the necessary massive computation to verify the 4-vex conjecture on the NeSI platform terminating on May 12, 2013. We also include the 189 patterns we conjecture characterize the multiplicity free Stanley symmetric functions below.

;; 2-vex= all perms whose stanley symmetric function has at most two terms, need to avoid all these patterns

(setf *2-vex* '((3 2 1 5 4) (2 1 5 4 3) (2 1 4 3 6 5) (2 4 1 3 6 5) (3 1 4 2 6 5) (3 1 2 6 4 5) (2 1 4 6 3 5) (2 4 1 6 3 5) (2 3 1 5 6 4) (2 1 5 3 6 4) (3 1 5 2 6 4) (4 2 6 1 5 3) (5 2 7 1 4 3 6) (5 1 7 3 2 6 4) (4 2 6 5 1 7 3) (2 5 4 7 1 6 3) (5 4 7 2 1 6 3) (5 2 7 6 1 4 3) (6 1 8 3 2 5 4 7) (2 6 4 8 1 5 3 7) (6 4 8 2 1 5 3 7) (2 6 5 8 1 4 3 7) (6 5 8 2 1 4 3 7) (5 1 7 3 6 2 8 4) (5 1 7 6 3 2 8 4) (6 1 8 3 7 2 5 4) (6 1 8 7 3 2 5 4) (2 5 4 7 6 1 8 3) (5 4 7 2 6 1 8 3) (5 4 7 6 2 1 8 3) (2 6 4 8 7 1 5 3) (6 4 8 7 2 1 5 3) (2 6 5 8 7 1 4 3) (6 5 8 2 7 1 4 3) (6 5 8 7 2 1 4 3)))

;; 3-vex= all perms whose stanley symmetric function has at most three terms, need to avoid all these patterns

(setf *3-vex* '((2 1 4 3 6 5) (4 3 2 1 6 5) (3 2 1 6 4 5) (4 2 1 6 3 5) (3 2 1 5 6 4) (3 2 5 1 6 4) (3 1 2 6 5 4) (2 3 1 6 5 4) (3 2 1 6 5 4) (3 1 6 2 5 4) (3 2 6 1 5 4) (2 4 1 6 5 3) (4 2 1 6 5 3) (4 2 6 1 5 3) (2 1 6 5 4 3) (3 5 2 1 4 7 6) (4 2 5 1 3 7 6) (2 5 4 1 3 7 6) (5 2 4 1 3 7 6) (4 3 1 5 2 7 6) (3 5 1 4 2 7 6) (5 3 1 4 2 7 6) (4 1 5 3 2 7 6) (3 5 2 4 1 7 6) (4 2 5 3 1 7 6) (2 4 1 3 7 5 6) (3 1 4 2 7 5 6) (3 1 2 5 7 4 6) (2 3 1 5 7 4 6) (2 5 1 3 7 4 6) (3 1 5 2 7 4 6) (3 5 1 2 7 4 6) (2 4 1 5 7 3 6) (2 5 1 4 7 3 6) (2 5 4 1 7 3 6) (2 1 5 7 4 3 6) (2 5 1 7 4 3 6) (5 2 7 1 4 3 6) (2 4 1 3 6 7 5) (3 1 4 2 6 7 5) (3 1 2 6 4 7 5) (2 3 1 6 4 7 5) (3 1 6 2 4 7 5) (2 4 1 6 3 7 5) (3 1 4 6 2 7 5) (3 4 1 6 2 7 5) (3 1 6 4 2 7 5) (4 1 6 3 2 7 5) (3 1 6 2 7 4 5) (2 4 1 6 7 3 5) (2 1 6 4 7 3 5) (2 1 4 7 6 3 5) (2 1 7 4 6 3 5) (2 1 6 5 3 7 4) (3 1 6 5 2 7 4) (2 1 5 7 3 6 4) (2 1 7 5 3 6 4) (5 1 7 3 2 6 4) (2 1 6 3 7 5 4) (4 2 6 5 1 7 3) (2 1 5 7 4 6 3) (2 5 4 7 1 6 3) (5 4 7 2 1 6 3) (2 1 6 4 7 5 3) (5 2 7 6 1 4 3) (2 4 6 1 3 5 8 7) (4 1 5 2 6 3 8 7) (4 1 2 3 8 5 6 7) (2 1 4 6 8 3 5 7) (2 4 6 1 8 3 5 7) (6 1 8 3 2 5 4 7) (2 6 4 8 1 5 3 7) (6 4 8 2 1 5 3 7) (2 6 5 8 1 4 3 7) (6 5 8 2 1 4 3 7) (3 4 1 2 7 8 5 6) (2 3 4 1 6 7 8 5) (2 1 6 3 7 4 8 5) (4 1 6 2 7 3 8 5) (5 1 7 3 6 2 8 4) (5 1 7 6 3 2 8 4) (6 1 8 3 7 2 5 4) (6 1 8 7 3 2 5 4) (2 5 4 7 6 1 8 3) (5 4 7 2 6 1 8 3) (5 4 7 6 2 1 8 3) (2 6 4 8 7 1 5 3) (6 4 8 7 2 1 5 3) (2 6 5 8 7 1 4 3) (6 5 8 2 7 1 4 3) (6 5 8 7 2 1 4 3)) )

(setf *4-vex* '((3 2 1 6 5 4) (3 2 6 1 5 4) (4 2 1 6 5 3) (2 1 4 3 6 7 5) (2 1 4 3 7 5 6) (2 1 4 3 7 6 5) (2 1 4 5 3 7 6) (2 1 4 6 3 7 5) (2 1 4 7 3 6 5) (2 1 5 3 4 7 6) (2 1 5 3 7 4 6) (2 1 5 3 7 6 4) (2 1 5 4 3 7 6) (2 1 5 4 7 3 6) (2 1 5 4 7 6 3) (2 1 6 4 3 7 5) (2 1 7 4 3 6 5) (2 1 7 6 5 4 3) (2 3 1 5 4 7 6) (2 3 1 5 7 6 4) (2 3 1 6 5 7 4) (2 3 1 6 7 5 4) (2 3 1 7 5 6 4) (2 3 1 7 6 5 4) (2 3 5 1 7 6 4) (2 4 1 3 7 6 5) (2 4 1 5 3 7 6) (2 4 1 6 3 7 5) (2 4 1 6 7 5 3) (2 4 1 7 3 6 5) (2 4 1 7 6 3 5) (2 4 1 7 6 5 3) (2 4 3 1 6 7 5) (2 5 1 3 7 6 4) (2 5 1 4 3 7 6) (2 5 1 4 7 3 6) (2 5 1 4 7 6 3) (2 5 1 7 3 6 4) (2 5 1 7 4 3 6) (2 5 1 7 4 6 3) (2 5 1 7 6 4 3) (2 5 4 1 7 3 6) (2 5 4 1 7 6 3) (3 1 2 5 4 7 6) (3 1 2 7 4 6 5) (3 1 2 7 5 4 6) (3 1 2 7 5 6 4) (3 1 2 7 6 4 5) (3 1 2 7 6 5 4) (3 1 4 2 7 6 5) (3 1 4 7 2 6 5) (3 1 5 2 4 7 6) (3 1 5 2 7 4 6) (3 1 5 2 7 6 4) (3 1 5 4 2 7 6) (3 1 5 7 2 6 4) (3 1 6 2 7 5 4) (3 1 6 4 2 7 5) (3 1 6 5 2 7 4) (3 1 7 2 6 4 5) (3 1 7 2 6 5 4) (3 1 7 4 2 6 5) (3 1 7 5 2 6 4) (3 1 7 6 2 5 4) (3 2 1 5 4 7 6) (3 2 1 5 7 4 6) (3 2 1 6 4 7 5) (3 2 1 6 7 4 5) (3 2 4 1 6 7 5) (3 2 5 1 4 7 6) (3 2 5 1 6 7 4) (3 2 5 1 7 4 6) (3 2 5 4 1 7 6) (3 2 6 1 4 7 5) (3 2 6 1 7 4 5) (3 2 6 4 1 7 5) (3 2 6 5 1 7 4) (3 4 1 2 7 6 5) (3 4 1 7 2 6 5) (3 4 2 1 6 7 5) (3 4 2 6 1 7 5) (3 5 1 2 7 6 4) (3 5 1 7 2 6 4) (3 5 2 1 7 4 6) (4 1 2 7 3 6 5) (4 1 3 2 7 5 6) (4 1 6 3 2 7 5) (4 1 7 3 2 6 5) (4 2 1 3 7 5 6) (4 2 1 5 3 7 6) (4 2 1 5 7 3 6) (4 2 1 6 3 7 5) (4 2 1 6 7 3 5) (4 2 1 7 3 5 6) (4 2 3 1 6 7 5) (4 2 3 1 7 5 6) (4 2 5 1 7 3 6) (4 2 6 1 3 7 5) (4 2 6 1 7 3 5) (4 2 6 3 1 7 5) (4 2 7 6 1 5 3) (4 3 1 2 7 5 6) (4 3 1 6 2 7 5) (4 3 2 1 6 7 5) (4 3 2 1 7 5 6) (4 3 2 6 1 7 5) (4 3 6 2 1 7 5) (5 2 1 4 3 7 6) (5 2 1 4 7 3 6) (5 2 1 7 4 3 6) (5 2 4 1 7 3 6) (5 2 7 1 6 4 3) (5 3 1 2 7 4 6) (5 3 2 1 7 4 6) (5 3 7 2 1 6 4) (5 4 2 1 7 3 6) (5 4 2 7 1 6 3) (5 4 3 2 1 7 6) (2 1 4 3 6 5 8 7) (2 1 4 8 7 6 3 5) (2 1 5 8 7 3 6 4) (2 1 5 8 7 4 3 6) (2 1 5 8 7 4 6 3) (2 1 6 8 3 7 5 4) (2 1 6 8 4 7 3 5) (2 1 6 8 4 7 5 3) (2 1 6 8 5 3 7 4) (2 1 6 8 5 4 3 7) (2 1 6 8 5 4 7 3) (2 1 6 8 5 7 4 3) (2 1 7 3 8 6 5 4) (2 1 7 4 8 6 3 5) (2 1 7 4 8 6 5 3) (2 1 7 5 8 3 6 4) (2 1 7 5 8 4 3 6) (2 1 7 5 8 4 6 3) (2 1 7 5 8 6 4 3) (2 1 7 6 3 8 5 4) (2 1 7 6 4 8 3 5) (2 1 7 6 4 8 5 3) (2 1 7 6 5 3 8 4) (2 1 8 4 7 6 3 5) (2 1 8 5 7 3 6 4) (2 1 8 5 7 4 3 6) (2 1 8 5 7 4 6 3) (2 1 8 6 3 7 5 4) (2 1 8 6 4 7 3 5) (2 1 8 6 4 7 5 3) (2 1 8 6 5 3 7 4) (2 1 8 7 4 6 3 5) (2 1 8 7 5 3 6 4) (2 3 1 5 6 8 4 7) (2 3 1 5 7 8 4 6) (2 3 1 6 4 7 8 5) (2 3 1 6 7 4 8 5) (2 3 5 1 4 7 8 6) (2 3 5 1 6 8 4 7) (2 3 5 1 7 8 4 6) (2 4 1 3 6 8 5 7) (2 4 1 3 7 5 8 6) (2 4 1 3 7 8 5 6) (2 4 1 5 7 8 3 6) (2 4 1 6 8 3 5 7) (2 4 1 7 5 8 3 6) (2 4 5 1 3 7 8 6) (2 4 5 1 7 8 3 6) (2 4 6 1 3 8 5 7) (2 5 1 3 4 8 6 7) (2 5 1 3 6 8 4 7) (2 5 1 3 7 8 4 6) (2 5 1 3 8 4 6 7) (2 5 1 7 6 3 8 4) (2 5 3 7 1 6 8 4) (2 5 3 8 1 6 7 4) (2 5 4 7 1 3 8 6) (2 5 4 8 7 1 6 3) (2 6 3 8 1 5 7 4) (2 6 4 8 1 3 7 5) (2 6 4 8 1 7 3 5) (2 6 4 8 1 7 5 3) (2 6 5 4 1 3 8 7) (2 6 5 4 8 1 7 3) (2 6 5 8 1 7 4 3) (3 1 2 5 8 4 6 7) (3 1 2 6 8 4 5 7) (3 1 2 7 4 5 8 6) (3 1 2 7 4 8 5 6) (3 1 4 2 6 8 5 7) (3 1 4 2 7 5 8 6) (3 1 4 2 7 8 5 6) (3 1 4 5 2 7 8 6) (3 1 4 6 2 7 8 5) (3 1 4 7 2 5 8 6) (3 1 4 7 2 8 5 6) (3 1 6 2 7 4 8 5) (3 1 6 8 2 5 4 7) (3 1 7 2 4 8 5 6) (3 1 7 2 5 8 4 6) (3 4 1 2 6 8 5 7) (3 4 1 2 7 5 8 6) (3 4 1 2 7 8 5 6) (3 4 1 5 2 7 8 6) (3 4 1 5 7 2 8 6) (3 4 1 6 2 7 8 5) (3 4 1 6 7 2 8 5) (3 4 1 7 2 5 8 6) (3 4 1 7 2 8 5 6) (3 5 1 2 4 8 6 7) (3 5 1 2 6 8 4 7) (3 5 1 2 7 8 4 6) (3 5 1 2 8 4 6 7) (3 5 1 4 7 2 8 6) (3 5 1 7 2 8 4 6) (3 6 1 2 4 8 5 7) (3 6 1 2 8 4 5 7) (3 6 1 4 2 8 5 7) (3 6 5 1 4 2 8 7) (3 6 5 2 1 4 8 7) (3 6 5 2 4 1 8 7) (3 6 5 8 2 1 7 4) (4 1 2 5 3 8 6 7) (4 1 2 7 3 5 8 6) (4 1 2 7 3 8 5 6) (4 1 5 2 3 8 6 7) (4 1 5 2 7 3 8 6) (4 1 7 2 3 8 5 6) (4 2 6 5 8 1 7 3) (4 2 6 8 1 5 3 7) (4 2 6 8 1 5 7 3) (4 2 6 8 5 1 7 3) (4 2 7 5 8 1 6 3) (4 2 7 6 5 1 8 3) (4 2 8 6 5 1 7 3) (4 5 2 7 8 1 6 3) (4 6 1 5 3 2 8 7) (4 6 2 5 1 3 8 7) (4 6 2 5 3 1 8 7) (4 6 2 5 8 1 7 3) (4 6 2 8 1 5 3 7) (4 6 2 8 1 5 7 3) (4 6 2 8 5 1 7 3) (4 6 3 1 5 2 8 7) (4 6 3 2 1 5 8 7) (4 6 3 2 5 1 8 7) (4 6 3 5 2 1 8 7) (5 1 3 8 2 6 4 7) (5 1 3 8 2 6 7 4) (5 1 3 8 6 2 7 4) (5 1 6 3 2 8 4 7) (5 1 6 3 8 2 7 4) (5 1 6 4 3 2 8 7) (5 1 7 3 8 2 6 4) (5 1 8 3 7 2 6 4) (5 1 8 7 3 2 6 4) (5 2 3 7 1 6 8 4) (5 2 3 8 1 6 4 7) (5 2 3 8 1 6 7 4) (5 2 3 8 6 1 7 4) (5 2 4 8 1 6 3 7) (5 2 4 8 1 6 7 3) (5 2 4 8 6 1 7 3) (5 2 6 3 8 1 7 4) (5 2 6 4 1 3 8 7) (5 2 6 4 3 1 8 7) (5 2 6 4 8 1 7 3) (5 2 7 1 6 3 8 4) (5 2 7 3 1 6 8 4) (5 2 7 3 8 1 6 4) (5 2 7 4 8 1 6 3) (5 2 7 6 1 8 4 3) (5 2 8 1 6 3 7 4) (5 2 8 3 1 6 7 4) (5 2 8 7 1 4 3 6) (5 2 8 7 6 1 4 3) (5 3 6 1 4 2 8 7) (5 3 6 2 1 4 8 7) (5 3 6 2 4 1 8 7) (5 3 6 4 2 1 8 7) (5 3 7 1 6 2 8 4) (5 3 7 2 6 1 8 4) (5 3 7 6 2 1 8 4) (5 3 8 1 6 2 7 4) (5 4 1 6 3 2 8 7) (5 4 2 6 1 3 8 7) (5 4 2 6 3 1 8 7) (5 4 2 7 6 1 8 3) (5 4 3 1 6 2 8 7) (5 4 8 7 2 1 6 3) (6 1 8 3 2 7 5 4) (6 1 8 4 3 2 7 5) (6 2 3 8 1 5 7 4) (6 2 4 8 1 3 7 5) (6 2 4 8 1 7 3 5) (6 2 5 4 1 3 8 7) (6 2 5 4 8 1 7 3) (6 2 8 1 4 3 7 5) (6 2 8 1 4 7 3 5) (6 2 8 1 5 3 7 4) (6 2 8 1 5 4 3 7) (6 2 8 1 5 4 7 3) (6 2 8 3 1 5 7 4) (6 2 8 4 1 3 7 5) (6 2 8 4 1 7 3 5) (6 2 8 4 3 1 7 5) (6 2 8 7 1 5 4 3) (6 3 5 1 4 2 8 7) (6 3 5 2 1 4 8 7) (6 3 5 2 4 1 8 7) (6 3 8 1 2 7 4 5) (6 3 8 1 4 2 7 5) (6 3 8 1 5 2 7 4) (6 3 8 2 1 5 4 7) (6 3 8 7 2 1 5 4) (6 4 1 5 3 2 8 7) (6 4 2 5 1 3 8 7) (6 4 2 5 3 1 8 7) (6 4 2 8 1 5 3 7) (6 4 3 1 5 2 8 7) (6 4 8 2 7 1 5 3) (6 5 1 8 3 2 7 4) (6 5 2 4 1 3 8 7) (6 5 2 8 1 4 3 7) (6 5 2 8 7 1 4 3) (6 5 3 1 4 2 8 7) (6 5 4 8 2 1 7 3) (6 5 8 2 1 7 4 3) (6 5 8 3 2 1 7 4) (2 1 4 6 9 8 3 5 7) (2 1 4 7 9 6 3 5 8) (2 1 4 7 9 6 8 3 5) (2 1 4 8 6 9 3 5 7) (2 1 4 8 6 9 7 3 5) (2 1 4 9 6 8 3 5 7) (2 1 5 7 9 3 6 4 8) (2 1 5 7 9 3 6 8 4) (2 1 5 7 9 4 3 6 8) (2 1 5 7 9 4 6 3 8) (2 1 5 7 9 4 6 8 3) (2 1 6 8 3 7 4 9 5) (2 1 6 8 5 7 3 9 4) (2 1 6 9 3 7 4 8 5) (2 1 7 3 8 4 9 6 5) (2 1 7 3 8 5 9 4 6) (2 1 7 3 8 5 9 6 4) (2 1 7 3 8 6 4 9 5) (2 1 7 4 8 5 9 3 6) (2 1 7 4 8 5 9 6 3) (2 1 7 5 8 6 3 9 4) (2 1 7 6 3 8 4 9 5) (2 1 7 9 4 6 3 5 8) (2 1 7 9 5 3 6 4 8) (2 1 8 3 9 5 7 4 6) (2 1 8 3 9 6 4 7 5) (2 1 8 4 6 9 3 5 7) (2 1 8 6 3 7 4 9 5) (2 1 9 4 6 8 3 5 7) (2 1 9 6 3 7 4 8 5) (2 3 6 5 8 1 7 9 4) (2 3 6 5 9 1 7 8 4) (2 3 7 5 9 1 6 8 4) (2 3 7 6 9 1 5 8 4) (2 4 7 6 1 3 5 9 8) (2 4 7 6 9 1 3 8 5) (2 4 7 6 9 1 8 3 5) (2 5 3 7 6 1 8 9 4) (2 5 4 7 6 9 1 8 3) (2 5 4 7 9 1 6 3 8) (2 5 4 7 9 1 6 8 3) (2 5 4 7 9 6 1 8 3) (2 5 4 8 6 9 1 7 3) (2 5 4 8 7 6 1 9 3) (2 5 4 9 7 6 1 8 3) (2 5 6 4 8 9 1 7 3) (2 5 7 4 1 3 6 9 8) (2 5 7 4 6 1 3 9 8) (2 5 7 4 6 9 1 8 3) (2 5 7 4 9 1 6 3 8) (2 5 7 4 9 1 6 8 3) (2 5 7 4 9 6 1 8 3) (2 6 3 8 7 1 5 9 4) (2 6 4 5 9 1 7 3 8) (2 6 4 5 9 1 7 8 3) (2 6 4 5 9 7 1 8 3) (2 6 4 7 1 3 5 9 8) (2 6 4 7 5 1 3 9 8) (2 6 4 7 5 9 1 8 3) (2 6 4 8 5 9 1 7 3) (2 6 4 8 7 1 3 9 5) (2 6 4 8 7 1 9 3 5) (2 6 4 8 7 1 9 5 3) (2 6 4 9 8 1 5 3 7) (2 6 4 9 8 7 1 5 3) (2 6 5 4 8 7 1 9 3) (2 6 5 8 1 7 3 9 4) (2 6 5 8 3 1 7 9 4) (2 6 5 8 7 1 9 4 3) (2 6 5 9 1 7 3 8 4) (2 6 5 9 3 1 7 8 4) (2 6 5 9 8 1 4 3 7) (2 6 5 9 8 7 1 4 3) (2 7 3 9 1 5 6 4 8) (2 7 3 9 8 1 5 6 4) (2 7 4 6 1 3 5 9 8) (2 7 4 9 1 3 6 5 8) (2 7 4 9 1 6 3 5 8) (2 7 4 9 1 6 5 3 8) (2 7 4 9 1 6 5 8 3) (2 7 4 9 8 1 3 6 5) (2 7 4 9 8 1 6 3 5) (2 7 4 9 8 1 6 5 3) (2 7 5 4 9 1 6 3 8) (2 7 5 9 1 6 3 8 4) (2 7 5 9 1 6 4 3 8) (2 7 5 9 1 6 4 8 3) (2 7 5 9 3 1 6 8 4) (2 7 5 9 4 8 1 6 3) (2 7 5 9 8 1 6 4 3) (2 7 6 4 9 1 5 3 8) (2 7 6 4 9 8 1 5 3) (2 7 6 5 9 1 4 3 8) (2 7 6 5 9 8 1 4 3) (2 7 6 9 1 4 3 8 5) (2 7 6 9 1 4 8 3 5) (2 7 6 9 1 5 3 8 4) (2 7 6 9 1 5 4 3 8) (2 7 6 9 1 5 4 8 3) (2 7 6 9 3 1 5 8 4) (2 7 6 9 4 1 3 8 5) (2 7 6 9 4 1 8 3 5) (2 7 6 9 8 1 5 4 3) (3 5 7 1 4 2 6 9 8) (3 5 7 1 4 6 2 9 8) (3 5 7 2 1 4 6 9 8) (3 5 7 2 4 1 6 9 8) (3 5 7 2 4 6 1 9 8) (3 6 5 8 1 7 2 9 4) (3 6 5 8 2 7 1 9 4) (3 6 5 8 7 2 1 9 4) (3 6 5 9 1 7 2 8 4) (3 7 5 9 1 6 2 8 4) (3 7 5 9 2 1 6 4 8) (3 7 5 9 8 2 1 6 4) (3 7 6 9 1 2 8 4 5) (3 7 6 9 1 4 2 8 5) (3 7 6 9 1 5 2 8 4) (3 7 6 9 2 1 5 4 8) (3 7 6 9 8 2 1 5 4) (4 2 6 5 8 7 1 9 3) (4 2 6 8 5 7 1 9 3) (4 2 7 5 8 6 1 9 3) (4 5 2 7 8 6 1 9 3) (4 6 1 5 2 7 3 9 8) (4 6 2 5 8 7 1 9 3) (4 6 2 8 5 7 1 9 3) (4 6 3 5 1 7 2 9 8) (4 7 1 5 2 6 3 9 8) (4 7 6 9 2 8 1 5 3) (5 1 3 8 6 7 2 9 4) (5 1 6 2 7 4 3 9 8) (5 1 6 3 7 2 4 9 8) (5 1 6 3 7 4 2 9 8) (5 1 6 3 8 7 2 9 4) (5 1 6 4 2 7 3 9 8) (5 1 7 3 8 6 2 9 4) (5 1 7 6 9 3 2 8 4) (5 1 7 9 3 2 6 4 8) (5 1 7 9 3 2 6 8 4) (5 1 7 9 3 6 2 8 4) (5 1 7 9 6 3 2 8 4) (5 1 8 3 7 6 2 9 4) (5 1 8 6 9 3 2 7 4) (5 1 8 7 3 6 2 9 4) (5 1 8 7 6 3 2 9 4) (5 1 9 3 7 6 2 8 4) (5 1 9 7 3 6 2 8 4) (5 1 9 7 6 3 2 8 4) (5 2 3 7 6 1 8 9 4) (5 2 3 8 6 7 1 9 4) (5 2 4 8 6 7 1 9 3) (5 2 6 3 7 1 4 9 8) (5 2 6 3 7 4 1 9 8) (5 2 6 3 8 7 1 9 4) (5 2 6 4 8 7 1 9 3) (5 2 7 3 6 1 8 9 4) (5 2 7 3 8 6 1 9 4) (5 2 7 4 8 6 1 9 3) (5 2 7 6 1 8 3 9 4) (5 2 7 6 3 1 8 9 4) (5 2 7 6 9 1 4 3 8) (5 2 7 6 9 8 1 4 3) (5 2 7 9 1 4 3 6 8) (5 2 7 9 6 1 4 3 8) (5 2 7 9 6 8 1 4 3) (5 2 8 6 9 1 4 3 7) (5 2 8 6 9 7 1 4 3) (5 3 6 4 1 7 2 9 8) (5 3 7 6 1 8 2 9 4) (5 4 1 6 2 7 3 9 8) (5 4 7 6 9 2 1 8 3) (5 4 7 9 2 1 6 3 8) (5 4 7 9 2 1 6 8 3) (5 4 7 9 2 6 1 8 3) (5 4 7 9 6 2 1 8 3) (5 4 8 6 9 2 1 7 3) (5 4 8 7 2 6 1 9 3) (5 4 8 7 6 2 1 9 3) (5 4 9 7 2 6 1 8 3) (5 4 9 7 6 2 1 8 3) (5 6 1 8 9 3 2 7 4) (5 6 2 8 9 1 4 3 7) (5 6 2 8 9 7 1 4 3) (5 6 4 8 9 2 1 7 3) (5 7 1 6 9 3 2 8 4) (5 7 1 9 3 2 6 4 8) (5 7 1 9 3 2 6 8 4) (5 7 1 9 3 6 2 8 4) (5 7 1 9 6 3 2 8 4) (5 7 2 4 1 3 6 9 8) (5 7 2 6 9 1 4 3 8) (5 7 2 6 9 8 1 4 3) (5 7 2 9 1 4 3 6 8) (5 7 2 9 6 1 4 3 8) (5 7 2 9 6 8 1 4 3) (5 7 3 1 4 2 6 9 8) (5 7 4 6 9 2 1 8 3) (5 7 4 9 2 1 6 3 8) (5 7 4 9 2 1 6 8 3) (5 7 4 9 2 6 1 8 3) (5 7 4 9 6 2 1 8 3) (6 1 2 9 4 3 7 5 8) (6 1 2 9 4 3 7 8 5) (6 1 2 9 4 7 3 8 5) (6 1 2 9 7 4 3 8 5) (6 1 3 9 2 5 4 7 8) (6 1 3 9 7 2 5 4 8) (6 1 3 9 7 8 2 5 4) (6 1 5 9 3 2 7 4 8) (6 1 5 9 3 2 7 8 4) (6 1 5 9 3 7 2 8 4) (6 1 5 9 7 3 2 8 4) (6 1 7 2 9 4 3 8 5) (6 1 7 3 5 2 4 9 8) (6 1 7 3 9 2 5 4 8) (6 1 7 3 9 8 2 5 4) (6 1 7 4 2 5 3 9 8) (6 1 7 5 9 3 2 8 4) (6 1 8 2 9 4 3 7 5) (6 1 8 3 2 7 4 9 5) (6 1 8 3 4 2 7 9 5) (6 1 8 3 7 2 9 5 4) (6 1 8 3 9 2 5 4 7) (6 1 8 3 9 7 2 5 4) (6 1 8 4 2 7 3 9 5) (6 1 8 4 3 7 2 9 5) (6 1 8 4 7 3 2 9 5) (6 1 8 5 9 3 2 7 4) (6 1 8 7 3 2 9 5 4) (6 1 8 7 4 3 2 9 5) (6 1 9 3 2 7 4 8 5) (6 1 9 3 4 2 7 8 5) (6 1 9 3 8 2 5 4 7) (6 1 9 3 8 7 2 5 4) (6 1 9 4 2 7 3 8 5) (6 1 9 8 3 2 5 4 7) (6 1 9 8 3 7 2 5 4) (6 1 9 8 7 3 2 5 4) (6 2 3 8 7 1 5 9 4) (6 2 3 9 1 5 4 7 8) (6 2 3 9 7 1 5 4 8) (6 2 3 9 7 8 1 5 4) (6 2 4 7 1 3 5 9 8) (6 2 4 8 7 1 3 9 5) (6 2 4 8 7 1 9 3 5) (6 2 4 9 1 5 3 7 8) (6 2 4 9 7 1 5 3 8) (6 2 4 9 7 8 1 5 3) (6 2 5 4 8 7 1 9 3) (6 2 5 9 1 4 3 7 8) (6 2 5 9 7 1 4 3 8) (6 2 5 9 7 8 1 4 3) (6 2 7 3 9 1 5 4 8) (6 2 7 3 9 8 1 5 4) (6 2 7 4 9 1 5 3 8) (6 2 7 4 9 8 1 5 3) (6 2 7 5 9 1 4 3 8) (6 2 7 5 9 8 1 4 3) (6 2 8 3 7 1 5 9 4) (6 2 8 3 9 1 5 4 7) (6 2 8 3 9 7 1 5 4) (6 2 8 4 3 7 1 9 5) (6 2 8 4 7 1 3 9 5) (6 2 8 4 7 1 9 3 5) (6 2 8 4 7 3 1 9 5) (6 2 8 4 9 1 5 3 7) (6 2 8 4 9 7 1 5 3) (6 2 8 5 9 1 4 3 7) (6 2 8 5 9 7 1 4 3) (6 2 8 7 1 4 3 9 5) (6 2 8 7 1 4 9 3 5) (6 2 8 7 1 5 3 9 4) (6 2 8 7 3 1 5 9 4) (6 2 8 7 4 1 3 9 5) (6 2 8 7 4 1 9 3 5) (6 2 8 7 4 3 1 9 5) (6 3 8 7 1 2 9 4 5) (6 3 8 7 1 4 2 9 5) (6 3 8 7 1 5 2 9 4) (6 4 1 5 2 7 3 9 8) (6 4 5 9 2 1 7 3 8) (6 4 5 9 2 1 7 8 3) (6 4 5 9 2 7 1 8 3) (6 4 5 9 7 2 1 8 3) (6 4 7 5 9 2 1 8 3) (6 4 8 2 7 5 1 9 3) (6 4 8 5 9 2 1 7 3) (6 4 8 7 2 9 1 5 3) (6 4 9 2 7 5 1 8 3) (6 4 9 8 2 1 5 3 7) (6 4 9 8 7 2 1 5 3) (6 5 1 8 3 7 2 9 4) (6 5 1 8 7 3 2 9 4) (6 5 4 8 2 7 1 9 3) (6 5 4 8 7 2 1 9 3) (6 5 8 2 1 7 3 9 4) (6 5 8 2 3 1 7 9 4) (6 5 8 2 7 1 9 4 3) (6 5 8 3 1 7 2 9 4) (6 5 8 3 2 7 1 9 4) (6 5 8 3 7 2 1 9 4) (6 5 8 7 2 1 9 4 3) (6 5 8 7 3 2 1 9 4) (6 5 9 2 1 7 3 8 4) (6 5 9 2 3 1 7 8 4) (6 5 9 3 1 7 2 8 4) (6 5 9 8 2 1 4 3 7) (6 5 9 8 2 7 1 4 3) (6 5 9 8 7 2 1 4 3) (7 1 9 3 2 5 4 8 6) (7 1 9 3 2 5 8 4 6) (7 1 9 3 2 6 4 8 5) (7 1 9 3 2 6 5 4 8) (7 1 9 3 2 6 5 8 4) (7 1 9 3 4 2 6 8 5) (7 1 9 3 5 2 4 8 6) 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;;;; When is a stanley symmetric function multiplicity free? We conjecture that these are all the patterns needed to characterize multiplicity free perms. We have shown this family respects patterns and have tested the conjecture through n=12.

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Sara Billey Last modified: Mon May 13 14:01:29 PDT 2013