A Computer-Generated Theorem by Shalosh B. Ekhad



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

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

              of L together, by identifying the last, 3, vertices

        of the previous link to the first, 3, vertices of the next link

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

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

                                                 a  b  c
                        times  the product of , x  y  z

                  where , a, b, c, are the last , 3, entries

                              of the P-partition



                    F[n](x, y, z), satisfies the following

                         functional-recurrence equation:

                        3  2                   3
                       q  x  y F[n - 1](1, 1, q  x y z)
F[n](x, y, z) = - -------------------------------------------
                    3  2      2                  3
                  (q  x  y - q  x y - q x + 1) (q  x y z - 1)

                             2
            q x F[n - 1](1, q  x y, q z)           F[n - 1](q x, q y, q z)
     + -------------------------------------- - -----------------------------
                   3  2      2                  (q z - 1) (q y - 1) (q x - 1)
       (q z - 1) (q  x  y - q  x y - q x + 1)

                               2
         q y F[n - 1](q x, 1, q  y z)
     + --------------------------------
                   2
       (q y - 1) (q  y z - 1) (q x - 1)



                        Subject to the intial condition

                       11  3  3  2    8  2  2      7  2  2      10  3  3
F[1](x, y, z) = (-1 - q   y  z  x  + q  y  z  x + q  y  z  x - q   z  y  x

        5  2        7  3  2      8  3  2      6  2  2      4  2
     + q  z  x y - q  z  y  x - q  z  y  x + q  y  z  x + q  z  y

        15  4  5  2    9  2  3      /              2          3
     + q   y  z  x  - q  y  z  x)  /  ((q z - 1) (q  z - 1) (q  z y - 1)
                                  /

      4            5              6              2            4
    (q  y z - 1) (q  x y z - 1) (q  x y z - 1) (q  y z - 1) (q  z x y - 1)

      3
    (q  x y z - 1))



              The pure generating function for all P-partitions is

                                 F[n](1, 1, 1)

                     This took, 1.544, seconds of CPU time