A Computer-Generated Theorem by Shalosh B. Ekhad



    Consider the poset L:= , [{}, {1}, {1}, {1, 2}, {1, 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:

                  4  2      6  2      5  2      7  3      12  4      7  2
F[n](x) = - (1 - q  x  - 3 q  x  - 2 q  x  + 2 q  x  - 3 q   x  - 2 q  x

        18  6      13  4      11  3    8  2      9  3    10  4      8  3
     + q   x  - 2 q   x  + 2 q   x  - q  x  + 4 q  x  - q   x  + 4 q  x

          10  3    14  4      11  4            5      /              2       2
     + 4 q   x  - q   x  - 2 q   x ) F[n - 1](q  x)  /  ((q x - 1) (q  x - 1)
                                                    /

      3       3   4       2   5
    (q  x - 1)  (q  x - 1)  (q  x - 1))



                        Subject to the intial condition

                4  2      6  2      5  2      7  3      12  4      7  2
F[1](x) = (1 - q  x  - 3 q  x  - 2 q  x  + 2 q  x  - 3 q   x  - 2 q  x

        18  6      13  4      11  3    8  2      9  3    10  4      8  3
     + q   x  - 2 q   x  + 2 q   x  - q  x  + 4 q  x  - q   x  + 4 q  x

          10  3    14  4      11  4    /              2       2   3       3
     + 4 q   x  - q   x  - 2 q   x )  /  ((q x - 1) (q  x - 1)  (q  x - 1)
                                     /

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



              The pure generating function for all P-partitions is

                                    F[n](1)

                     This took, 1.676, seconds of CPU time