A Computer-Generated Theorem by Shalosh B. Ekhad



 Consider the poset L:= , [{}, {1}, {1}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3, 4, 5}]

             Let P[n] be  the poset obtained from joining n copies

              of L together, by identifying the last, 1, vertices

        of the previous link to the first, 1, vertices of the next link

                  Let , F[n](x),  be the weight-enumerator of

        of all P-partitions of P[n], with the weight q^(sum of entries)

                                                    a
                           times  the product of , x

                     where , a, are the last , 1, entries

                              of the P-partition



                       F[n](x), satisfies the following

                         functional-recurrence equation:

                          12  4    8  2    4  2                5
                        (q   x  - q  x  - q  x  + 1) F[n - 1](q  x)
      F[n](x) = - -------------------------------------------------------
                              2       2   3          4       2   5
                  (q x - 1) (q  x - 1)  (q  x - 1) (q  x - 1)  (q  x - 1)



                        Subject to the intial condition

                                 12  4    8  2    4  2
                                q   x  - q  x  - q  x  + 1
  F[1](x) = ------------------------------------------------------------------
                        2       2   3          4       2   5          6
            (q x - 1) (q  x - 1)  (q  x - 1) (q  x - 1)  (q  x - 1) (q  x - 1)



              The pure generating function for all P-partitions is

                                    F[n](1)

                     This took, 1.140, seconds of CPU time