Support for Proofs in Foata and Zeilberger's article

                Babson and Steingrimsson's Newly Discovered ...

                    In Markovian Statistics language, maj is

                              [0, -1 + n, j, i, n]

                    In Markovian Statistics language, des is

                                [0, 1, j, i, n]

                   In Markovian Statistics language, rise is

                                [1, 0, j, i, n]

        Stuff for Proof of Theorem 1(the second part of Conj. 8 of [BS])

                    In Markovian Statistics langauge, S11 is

                           [0, -i - 2 + 2 j, j, i, n]

                         Its  Umbral Operator  is F(z)->

                          n                            2
                    F(1) z  z   F(z) z   F(z q) z   F(q ) z
                    --------- - ------ - -------- + -------
                      z - 1     z - 1     -z + q    -z + q

                                                (-1 + n)    (-1 + n)
                                            z (z         - q        )
           When you apply this operator to, -------------------------
                                                      z - q

                                    You get

                                n    n        n
                              (z  - q ) (q - q ) z
                              --------------------
                              q (-1 + q) (-z + q)

        Stuff for Proof of Theorem 2(the third part of Conj. 8 of [BS])

                    In Markovian Statistics langauge, S13 is

                        [2 i - 2 j - 1, i - 1, j, i, n]

                            Its  Umbral Operator  is

                         Its  Umbral Operator  is F(z)->

                 1        n 2  n
              F(----) q (q )  z  z
                  2
                 q                   F(1) z    F(z q) z   F(z) z q
              -------------------- - ------- + -------- - --------
                       2             z q - 1   z q - 1       2
                    z q  - 1                              z q  - 1

                                                   (-1 + n)  (-1 + n)
                                           z (1 - z         q        )
          When you apply this operator to, ---------------------------
                                                    -z q + 1

                                    You get

                                    n  n        n
                             (-1 + z  q ) (q - q ) z
                           - -----------------------
                              q (-1 + q) (z q - 1)

             Stuff for the proof of Theorem 3 of Foata-Zeilberger

      The Umbral operator linking A_{n-1} to A_n, for [S5,rise] is F(z)->

                                 n                 n
                      F(q) z (t z  - 1)   F(z) z (q  t - 1)
                    - ----------------- + -----------------
                           -z + q              -z + q

      The Umbral operator linking A_{n-1} to A_n, for [maj,des] is F(z)->

                          n  n    n                  n  n
                F(1) z (-q  t  + z  q t)   F(z) z (-q  t  + q t)
                ------------------------ - ---------------------
                      (z - 1) q t               (z - 1) q t

                 The whole thing took, .750, seconds of CPU time