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, 2, vertices

        of the previous link to the first, 2, vertices of the next link

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

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

                                                  a  b
                         times  the product of , x  y

                    where , a, b, are the last , 2, entries

                              of the P-partition



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

                         functional-recurrence equation:

                         4  2                     3
                       (q  y  x - 1) F[n - 1](q, q  x y)
F[n](x, y) = ------------------------------------------------------
               2                    3                      2
             (q  x y - 1) (q - 1) (q  x y - 1) (q y - 1) (q  y - 1)

                  4  2                       4
                (q  y  x - 1) q F[n - 1](1, q  x y)
     - ------------------------------------------------------
         4            2                              2
       (q  x y - 1) (q  x y - 1) (q - 1) (q y - 1) (q  y - 1)



                        Subject to the intial condition

               12  3  4    8  2  2    4  2          /              2
F[1](x, y) = (q   x  y  - q  x  y  - q  y  x + 1)  /  ((q y - 1) (q  y - 1)
                                                  /

      3            4         2   5            6            2
    (q  x y - 1) (q  x y - 1)  (q  x y - 1) (q  x y - 1) (q  x y - 1))



              The pure generating function for all P-partitions is

                                   F[n](1, 1)

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