A Computer-Generated Theorem by Shalosh B. Ekhad



Consider the poset L:= ,

    [{}, {1}, {1, 2}, {1, 2}, {1, 2, 3, 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:

F[n](x, y, z) =

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



                        Subject to the intial condition

                  6  2  2          /              2            3
F[1](x, y, z) = (q  y  z  x - 1)  /  ((q z - 1) (q  y z - 1) (q  y z - 1)
                                 /

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



              The pure generating function for all P-partitions is

                                 F[n](1, 1, 1)

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