Conjectured Explicit Formulas Enumerating 4-rowed Truncated Standard Young \

    Tableau of shape [n,n+a1,n+a2,n+a3] for a from 0<=a1,a2,a3<=, 5,

    and a3<a2  divided by the number syt of shape, [n, n, n, n]



                              By Shalosh B. Ekhad





Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n],  is  the number of syt of [n$4]



                        8 (4 n + 1) (2 n + 1) (4 n + 3)
                        -------------------------------
                        3 (3 n + 2) (3 n + 1) (4 + 3 n)



                             and in Maple notation



8/3*(4*n+1)*(2*n+1)*(4*n+3)/(3*n+2)/(3*n+1)/(4+3*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n],  is  the number of syt of [n$4]



                   8 (7 n + 5) (4 n + 1) (2 n + 1) (4 n + 3)
                   -----------------------------------------
                   3 (3 n + 2) (3 n + 5) (3 n + 1) (4 + 3 n)



                             and in Maple notation



8/3*(7*n+5)*(4*n+1)*(2*n+1)*(4*n+3)/(3*n+2)/(3*n+5)/(3*n+1)/(4+3*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, 1 + n],  is  the number of syt of [n$4]



                   80 (1 + n) (4 n + 3) (2 n + 1) (4 n + 1)
                   -----------------------------------------
                   3 (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



80/3*(1+n)*(4*n+3)*(2*n+1)*(4*n+1)/(4+3*n)/(3*n+1)/(3*n+5)/(3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n],  is  the number of syt of [n$4]



                                                   2
              8 (4 n + 3) (2 n + 1) (4 n + 1) (43 n  + 71 n + 30)
              ---------------------------------------------------
               9 (n + 2) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(43*n^2+71*n+30)/(n+2)/(4+3*n)/(3*n+1)/(3*n+5)/(3*n
+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, 1 + n],  is  the number of syt of [n$4]



                                                   2
              8 (4 n + 3) (2 n + 1) (4 n + 1) (73 n  + 161 n + 90)
              ----------------------------------------------------
               9 (n + 2) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(73*n^2+161*n+90)/(n+2)/(4+3*n)/(3*n+1)/(3*n+5)/(3*
n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 2],  is  the number of syt of [n$4]



                                                   2
             8 (4 n + 3) (2 n + 1) (4 n + 1) (103 n  + 251 n + 150)
             ------------------------------------------------------
               9 (n + 2) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(103*n^2+251*n+150)/(n+2)/(4+3*n)/(3*n+1)/(3*n+5)/(
3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n],  is  the number of syt of [n$4]



                                                        2
          40 (1 + n) (4 n + 3) (2 n + 1) (4 n + 1) (49 n  + 89 n + 42)
          ------------------------------------------------------------
          9 (3 n + 7) (n + 2) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



40/9*(1+n)*(4*n+3)*(2*n+1)*(4*n+1)*(49*n^2+89*n+42)/(3*n+7)/(n+2)/(4+3*n)/(3*n+
1)/(3*n+5)/(3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, 1 + n],  is  the number of syt of [n$4]



                 2
         32 (29 n  + 69 n + 42) (4 n + 1) (2 n + 1) (4 n + 3) (4 n + 5)
         --------------------------------------------------------------
          9 (n + 2) (3 n + 7) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



32/9*(29*n^2+69*n+42)*(4*n+1)*(2*n+1)*(4*n+3)*(4*n+5)/(n+2)/(3*n+7)/(4+3*n)/(3*
n+1)/(3*n+5)/(3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 2],  is  the number of syt of [n$4]



                                             3         2
       8 (4 n + 3) (2 n + 1) (4 n + 1) (773 n  + 3158 n  + 4259 n + 1890)
       ------------------------------------------------------------------
          9 (n + 2) (3 n + 7) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(773*n^3+3158*n^2+4259*n+1890)/(n+2)/(3*n+7)/(4+3*n
)/(3*n+1)/(3*n+5)/(3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 3],  is  the number of syt of [n$4]



                                             3         2
      16 (4 n + 3) (2 n + 1) (4 n + 1) (541 n  + 2316 n  + 3233 n + 1470)
      -------------------------------------------------------------------
          9 (n + 2) (3 n + 7) (4 + 3 n) (3 n + 1) (3 n + 5) (3 n + 2)



                             and in Maple notation



16/9*(4*n+3)*(2*n+1)*(4*n+1)*(541*n^3+2316*n^2+3233*n+1470)/(n+2)/(3*n+7)/(4+3*
n)/(3*n+1)/(3*n+5)/(3*n+2)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n],  is  the number of syt of [n$4]



            2
    32 (83 n  + 163 n + 84) (4 n + 1) (2 n + 1) (4 n + 3) (1 + n) (4 n + 5)
    -----------------------------------------------------------------------
     9 (n + 2) (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 1) (4 + 3 n) (3 n + 7)



                             and in Maple notation



32/9*(83*n^2+163*n+84)*(4*n+1)*(2*n+1)*(4*n+3)*(1+n)*(4*n+5)/(n+2)/(3*n+2)/(3*n
+5)/(3*n+8)/(3*n+1)/(4+3*n)/(3*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, 1 + n],  is  the number of syt of [n$4]



            2
   160 (17 n  + 43 n + 28) (4 n + 1) (2 n + 1) (4 n + 3) (4 n + 5) (2 n + 3)
   -------------------------------------------------------------------------
     9 (n + 2) (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 1) (4 + 3 n) (3 n + 7)



                             and in Maple notation



160/9*(17*n^2+43*n+28)*(4*n+1)*(2*n+1)*(4*n+3)*(4*n+5)*(2*n+3)/(n+2)/(3*n+2)/(3
*n+5)/(3*n+8)/(3*n+1)/(4+3*n)/(3*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 2],  is  the number of syt of [n$4]



8 (4 n + 3) (2 n + 1) (4 n + 1)

           4          3          2
    (5039 n  + 30018 n  + 66541 n  + 64962 n + 23520)/(9 (n + 2) (3 n + 8)

    (3 n + 5) (3 n + 2) (3 n + 7) (4 + 3 n) (3 n + 1))



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(5039*n^4+30018*n^3+66541*n^2+64962*n+23520)/(n+2)/
(3*n+8)/(3*n+5)/(3*n+2)/(3*n+7)/(4+3*n)/(3*n+1)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 3],  is  the number of syt of [n$4]



               3         2
     40 (1657 n  + 7200 n  + 10199 n + 4704) (4 n + 1) (2 n + 1) (4 n + 3)
     ---------------------------------------------------------------------
         9 (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 1) (4 + 3 n) (3 n + 7)



                             and in Maple notation



40/9*(1657*n^3+7200*n^2+10199*n+4704)*(4*n+1)*(2*n+1)*(4*n+3)/(3*n+2)/(3*n+5)/(
3*n+8)/(3*n+1)/(4+3*n)/(3*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 4],  is  the number of syt of [n$4]



8 (4 n + 3) (2 n + 1) (4 n + 1)

            4          3           2
    (11531 n  + 75122 n  + 179449 n  + 186058 n + 70560)/(9 (n + 2) (3 n + 8)

    (3 n + 5) (3 n + 2) (3 n + 7) (4 + 3 n) (3 n + 1))



                             and in Maple notation



8/9*(4*n+3)*(2*n+1)*(4*n+1)*(11531*n^4+75122*n^3+179449*n^2+186058*n+70560)/(n+
2)/(3*n+8)/(3*n+5)/(3*n+2)/(3*n+7)/(4+3*n)/(3*n+1)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n],  is  the number of syt of [n$4]



                             5 (4 n + 3) (4 n + 1)
                             ---------------------
                              (5 + 2 n) (2 n + 3)



                             and in Maple notation



5*(4*n+3)*(4*n+1)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n],  is  the number of syt of [n$4]



                        (4 n + 3) (4 n + 1) (11 n + 10)
                        -------------------------------
                          (n + 2) (5 + 2 n) (2 n + 3)



                             and in Maple notation



(4*n+3)*(4*n+1)*(11*n+10)/(n+2)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, 1 + n],  is  the number of syt of [n$4]



                        4 (4 n + 1) (4 n + 3) (4 n + 5)
                        -------------------------------
                          (n + 2) (5 + 2 n) (2 n + 3)



                             and in Maple notation



4*(4*n+1)*(4*n+3)*(4*n+5)/(n+2)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n],  is  the number of syt of [n$4]



                                                        2
          40 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (11 n  + 31 n + 21)
          ------------------------------------------------------------
          3 (3 n + 7) (4 + 3 n) (n + 2) (5 + 2 n) (2 n + 3) (3 n + 5)



                             and in Maple notation



40/3*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(11*n^2+31*n+21)/(3*n+7)/(4+3*n)/(n+2)/(5+2*
n)/(2*n+3)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, 1 + n],  is  the number of syt of [n$4]



                3        2
        4 (191 n  + 852 n  + 1267 n + 630) (4 n + 1) (4 n + 3) (4 n + 5)
        ----------------------------------------------------------------
          3 (n + 2) (3 n + 7) (4 + 3 n) (5 + 2 n) (2 n + 3) (3 n + 5)



                             and in Maple notation



4/3*(191*n^3+852*n^2+1267*n+630)*(4*n+1)*(4*n+3)*(4*n+5)/(n+2)/(3*n+7)/(4+3*n)/
(5+2*n)/(2*n+3)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 2],  is  the number of syt of [n$4]



             8 (17 n + 25) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
             -----------------------------------------------------
               3 (n + 2) (3 n + 7) (4 + 3 n) (5 + 2 n) (3 n + 5)



                             and in Maple notation



8/3*(17*n+25)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(3*n+7)/(4+3*n)/(5+2*n)/(3*
n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n],  is  the number of syt of [n$4]



                                              3         2
4 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (611 n  + 2577 n  + 3602 n + 1680)/(3

    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (5 + 2 n) (2 n + 3))



                             and in Maple notation



4/3*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(611*n^3+2577*n^2+3602*n+1680)/(3*n+7)/(4+3*n
)/(3*n+8)/(3*n+5)/(n+2)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, 1 + n],  is  the number of syt of [n$4]



                2
        16 (37 n  + 107 n + 80) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
        ---------------------------------------------------------------
          3 (n + 2) (3 n + 8) (3 n + 7) (4 + 3 n) (5 + 2 n) (3 n + 5)



                             and in Maple notation



16/3*(37*n^2+107*n+80)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(3*n+8)/(3*n+7)/(4
+3*n)/(5+2*n)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 2],  is  the number of syt of [n$4]



         40 (5 n + 9) (5 n + 8) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
         --------------------------------------------------------------
          3 (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n)



                             and in Maple notation



40/3*(5*n+9)*(5*n+8)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(3*n+7)/(4+3*n)/(3*n
+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 3],  is  the number of syt of [n$4]



                2
        32 (44 n  + 159 n + 140) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
        ----------------------------------------------------------------
          3 (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n)



                             and in Maple notation



32/3*(44*n^2+159*n+140)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(3*n+7)/(4+3*n)/(
3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n],  is  the number of syt of [n$4]



                                                            2
    8 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (203 n  + 529 n + 360)
    ------------------------------------------------------------------------
      9 (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3) (n + 2) (5 + 2 n)



                             and in Maple notation



8/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(203*n^2+529*n+360)/(3*n+7)/(4+3*n)/(
3*n+8)/(3*n+5)/(n+3)/(n+2)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, 1 + n],  is  the number of syt of [n$4]



                      2
    40 (5 n + 9) (17 n  + 51 n + 40) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
    ------------------------------------------------------------------------
      9 (n + 3) (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n)



                             and in Maple notation



40/9*(5*n+9)*(17*n^2+51*n+40)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+3)/(n+2)/(3*n+
7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 2],  is  the number of syt of [n$4]



                                                       2
            320 (4 n + 5) (4 n + 3) (4 n + 1) (4 n + 7)  (5 n + 9)
          -----------------------------------------------------------
          9 (n + 3) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n)



                             and in Maple notation



320/9*(4*n+5)*(4*n+3)*(4*n+1)*(4*n+7)^2*(5*n+9)/(n+3)/(3*n+7)/(4+3*n)/(3*n+8)/(
3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 3],  is  the number of syt of [n$4]



        2
32 (83 n  + 309 n + 280) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7) (4 n + 9)/(9

    (n + 3) (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n))



                             and in Maple notation



32/9*(83*n^2+309*n+280)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)*(4*n+9)/(n+3)/(n+2)/(3*
n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 4],  is  the number of syt of [n$4]



                2
       128 (58 n  + 212 n + 189) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
       -----------------------------------------------------------------
           9 (n + 3) (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5)



                             and in Maple notation



128/9*(58*n^2+212*n+189)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+3)/(n+2)/(3*n+7)/(4
+3*n)/(3*n+8)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n],  is  the number of syt of [n$4]



                                              2
                   3 (4 n + 3) (4 n + 1) (11 n  + 33 n + 20)
                   -----------------------------------------
                      (n + 3) (n + 2) (5 + 2 n) (2 n + 3)



                             and in Maple notation



3*(4*n+3)*(4*n+1)*(11*n^2+33*n+20)/(n+3)/(n+2)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n],  is  the number of syt of [n$4]



                                     4         3         2
10 (1 + n) (4 n + 3) (4 n + 1) (553 n  + 3373 n  + 7507 n  + 7215 n + 2520)/(3

    (3 n + 7) (4 + 3 n) (n + 3) (n + 2) (5 + 2 n) (2 n + 3) (3 n + 5))



                             and in Maple notation



10/3*(1+n)*(4*n+3)*(4*n+1)*(553*n^4+3373*n^3+7507*n^2+7215*n+2520)/(3*n+7)/(4+3
*n)/(n+3)/(n+2)/(5+2*n)/(2*n+3)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, 1 + n],  is  the number of syt of [n$4]



               3         2
       4 (269 n  + 1314 n  + 2071 n + 1050) (4 n + 1) (4 n + 3) (4 n + 5)
       ------------------------------------------------------------------
           3 (n + 3) (n + 2) (3 n + 7) (4 + 3 n) (5 + 2 n) (3 n + 5)



                             and in Maple notation



4/3*(269*n^3+1314*n^2+2071*n+1050)*(4*n+1)*(4*n+3)*(4*n+5)/(n+3)/(n+2)/(3*n+7)/
(4+3*n)/(5+2*n)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n],  is  the number of syt of [n$4]



                                              3         2
8 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (493 n  + 2328 n  + 3507 n + 1680)/(3

    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3) (n + 2) (5 + 2 n))



                             and in Maple notation



8/3*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(493*n^3+2328*n^2+3507*n+1680)/(3*n+7)/(4+3*n
)/(3*n+8)/(3*n+5)/(n+3)/(n+2)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, 1 + n],  is  the number of syt of [n$4]



         4          3          2
4 (1793 n  + 11736 n  + 28395 n  + 30092 n + 11760) (4 n + 1) (4 n + 3)

    (4 n + 5)/(3 (n + 3) (n + 2) (3 n + 8) (3 n + 7) (4 + 3 n) (5 + 2 n)

    (3 n + 5))



                             and in Maple notation



4/3*(1793*n^4+11736*n^3+28395*n^2+30092*n+11760)*(4*n+1)*(4*n+3)*(4*n+5)/(n+3)/
(n+2)/(3*n+8)/(3*n+7)/(4+3*n)/(5+2*n)/(3*n+5)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 2],  is  the number of syt of [n$4]



                      2
    40 (5 n + 9) (13 n  + 43 n + 32) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)
    ------------------------------------------------------------------------
      3 (n + 3) (n + 2) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n)



                             and in Maple notation



40/3*(5*n+9)*(13*n^2+43*n+32)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+3)/(n+2)/(3*n+
7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n],  is  the number of syt of [n$4]



4 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

           4          3          2                      /
    (5333 n  + 33852 n  + 79081 n  + 80646 n + 30240)  /  (9 (3 n + 7)
                                                      /

                                                 2
    (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (n + 3)  (5 + 2 n))



                             and in Maple notation



4/9*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(5333*n^4+33852*n^3+79081*n^2+80646*n+30240)/
(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+2)/(n+3)^2/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, 1 + n],  is  the number of syt of [n$4]



         4         3          2
8 (1339 n  + 8973 n  + 22265 n  + 24231 n + 9720) (4 n + 1) (4 n + 3) (4 n + 5)

                /                   2
    (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5)
              /

    (5 + 2 n))



                             and in Maple notation



8/9*(1339*n^4+8973*n^3+22265*n^2+24231*n+9720)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/
(n+2)/(n+3)^2/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 2],  is  the number of syt of [n$4]



          4         3          2
16 (1157 n  + 8439 n  + 22705 n  + 26583 n + 11340) (4 n + 1) (4 n + 3)

                          /                   2
    (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 7) (4 + 3 n) (3 n + 8)
                        /

    (3 n + 5) (5 + 2 n))



                             and in Maple notation



16/9*(1157*n^4+8439*n^3+22705*n^2+26583*n+11340)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7
)/(n+2)/(n+3)^2/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 3],  is  the number of syt of [n$4]



         4          3          2
8 (3289 n  + 24783 n  + 68555 n  + 82101 n + 35640) (4 n + 1) (4 n + 3)

                          /                   2
    (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 7) (4 + 3 n) (3 n + 8)
                        /

    (3 n + 5) (5 + 2 n))



                             and in Maple notation



8/9*(3289*n^4+24783*n^3+68555*n^2+82101*n+35640)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7
)/(n+2)/(n+3)^2/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n],  is  the number of syt of [n$4]



8 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

           4          3          2                      /
    (3473 n  + 22468 n  + 53637 n  + 56070 n + 21600)  /  (9 (3 n + 10)
                                                      /

                                                           2
    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (n + 3)  (5 + 2 n))



                             and in Maple notation



8/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(3473*n^4+22468*n^3+53637*n^2+56070*n
+21600)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+2)/(n+3)^2/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, 1 + n],  is  the number of syt of [n$4]



80 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

          4         3          2                     /
    (749 n  + 5127 n  + 13009 n  + 14487 n + 5940)  /  (9 (3 n + 10) (3 n + 7)
                                                   /

                                         2
    (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3)  (5 + 2 n))



                             and in Maple notation



80/9*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(749*n^4+5127*n^3+13009*n^2+14487*n+5940)/
(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+3)^2/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 2],  is  the number of syt of [n$4]



64 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

          4         3         2                     /
    (451 n  + 3361 n  + 9239 n  + 11045 n + 4800)  /  (9 (n + 2) (3 n + 10)
                                                  /

                                                   2
    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3)  (5 + 2 n))



                             and in Maple notation



64/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(451*n^4+3361*n^3+9239*n^2+11045*n
+4800)/(n+2)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+3)^2/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 3],  is  the number of syt of [n$4]



          5           4           3            2
8 (24299 n  + 247263 n  + 991135 n  + 1950501 n  + 1876770 n + 702000)

                                              /
    (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)  /  (9 (n + 2) (3 n + 10) (3 n + 7)
                                            /

                                         2
    (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3)  (5 + 2 n))



                             and in Maple notation



8/9*(24299*n^5+247263*n^4+991135*n^3+1950501*n^2+1876770*n+702000)*(4*n+1)*(4*n
+3)*(4*n+5)*(4*n+7)/(n+2)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+3)^2/(5+2
*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 4],  is  the number of syt of [n$4]



           5           4           3            2
16 (17083 n  + 177251 n  + 722315 n  + 1441177 n  + 1402350 n + 529200)

                                              /                   2
    (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 10)
                                            /

    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (5 + 2 n))



                             and in Maple notation



16/9*(17083*n^5+177251*n^4+722315*n^3+1441177*n^2+1402350*n+529200)*(4*n+1)*(4*
n+3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(5+
2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n],  is  the number of syt of [n$4]



                                                   2
               30 (1 + n) (4 n + 3) (4 n + 1) (10 n  + 32 n + 21)
               --------------------------------------------------
                 (n + 3) (n + 2) (2 n + 7) (5 + 2 n) (2 n + 3)



                             and in Maple notation



30*(1+n)*(4*n+3)*(4*n+1)*(10*n^2+32*n+21)/(n+3)/(n+2)/(2*n+7)/(5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n],  is  the number of syt of [n$4]



2 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

            5           4           3            2
    (25987 n  + 231872 n  + 804421 n  + 1355828 n  + 1109892 n + 352800)/(3

    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3) (n + 2) (2 n + 7) (5 + 2 n)

    (2 n + 3))



                             and in Maple notation



2/3*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(25987*n^5+231872*n^4+804421*n^3+1355828*n^2+
1109892*n+352800)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+3)/(n+2)/(2*n+7)/(5+2*n)/(
2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, 1 + n],  is  the number of syt of [n$4]



         5          4           3           2
8 (5483 n  + 50509 n  + 182111 n  + 321407 n  + 277802 n + 94080) (4 n + 1)

    (4 n + 3) (4 n + 5)/(3 (n + 3) (n + 2) (3 n + 8) (3 n + 5) (3 n + 7)

    (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



8/3*(5483*n^5+50509*n^4+182111*n^3+321407*n^2+277802*n+94080)*(4*n+1)*(4*n+3)*(
4*n+5)/(n+3)/(n+2)/(3*n+8)/(3*n+5)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n],  is  the number of syt of [n$4]



4 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

            5           4            3            2                         /
    (35794 n  + 322625 n  + 1133954 n  + 1943101 n  + 1623678 n + 529200)  /  (
                                                                          /

                                                             2
    9 (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (n + 3)  (2 n + 7)

    (5 + 2 n))



                             and in Maple notation



4/9*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(35794*n^5+322625*n^4+1133954*n^3+1943101*n^2
+1623678*n+529200)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+2)/(n+3)^2/(2*n+7)/(5+2*n
)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, 1 + n],  is  the number of syt of [n$4]



          5           4           3            2
4 (17173 n  + 159989 n  + 584621 n  + 1047787 n  + 921102 n + 317520) (4 n + 1)

                                    /                   2
    (4 n + 3) (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 8) (3 n + 5)
                                  /

    (3 n + 7) (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



4/9*(17173*n^5+159989*n^4+584621*n^3+1047787*n^2+921102*n+317520)*(4*n+1)*(4*n+
3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+8)/(3*n+5)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n
)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 2],  is  the number of syt of [n$4]



          5          4           3           2
16 (6652 n  + 65231 n  + 251414 n  + 475303 n  + 439728 n + 158760) (4 n + 1)

                                    /                   2
    (4 n + 3) (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 8) (3 n + 5)
                                  /

    (3 n + 7) (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/9*(6652*n^5+65231*n^4+251414*n^3+475303*n^2+439728*n+158760)*(4*n+1)*(4*n+3)
*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+8)/(3*n+5)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, n],  is  the number of syt of [n$4]



4 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

            5           4            3            2                         /
    (47201 n  + 430262 n  + 1532585 n  + 2668044 n  + 2271060 n + 756000)  /  (
                                                                          /

                                                                        2
    9 (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (n + 3)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



4/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(47201*n^5+430262*n^4+1532585*n^3+
2668044*n^2+2271060*n+756000)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+2)/(n
+3)^2/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, 1 + n],  is  the number of syt of [n$4]



160 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

           5          4          3           2                       /
    (2468 n  + 23293 n  + 86329 n  + 157097 n  + 140313 n + 49140)  /  (9
                                                                   /

                                                              2
    (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3)  (2 n + 7)

    (5 + 2 n))



                             and in Maple notation



160/9*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(2468*n^5+23293*n^4+86329*n^3+157097*n^2+
140313*n+49140)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+3)^2/(2*n+7)/(5+2*n
)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, n + 2],  is  the number of syt of [n$4]



16 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

            5           4           3           2                        /
    (11159 n  + 111018 n  + 434135 n  + 832596 n  + 780980 n + 285600)  /  (9
                                                                       /

                   2
    (n + 2) (n + 3)  (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (4 + 3 n)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(11159*n^5+111018*n^4+434135*n^3+
832596*n^2+780980*n+285600)/(n+2)/(n+3)^2/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3
*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, n + 3],  is  the number of syt of [n$4]



           6           5            4            3            2
32 (32296 n  + 403358 n  + 2071127 n  + 5588824 n  + 8344749 n  + 6522390 n

                                                          /                   2
     + 2079000) (4 n + 1) (4 n + 3) (4 n + 5) (4 n + 7)  /  (9 (n + 2) (n + 3)
                                                        /

    (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



32/9*(32296*n^6+403358*n^5+2071127*n^4+5588824*n^3+8344749*n^2+6522390*n+
2079000)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+8)/(3*n+5)/(3*n+10)
/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n],  is  the number of syt of [n$4]



16 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

            5           4            3            2                          /
    (60308 n  + 556021 n  + 2006250 n  + 3544347 n  + 3067110 n + 1039500)  /
                                                                           /

                                                                            2
    (9 (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(60308*n^5+556021*n^4+2006250*n^3+
3544347*n^2+3067110*n+1039500)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5
)/(n+3)^2/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, 1 + n],  is  the number of syt of [n$4]



16 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

            5           4            3            2                         /
    (33587 n  + 321079 n  + 1206165 n  + 2226053 n  + 2016760 n + 716100)  /  (
                                                                          /

                                                                           2
    9 (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(33587*n^5+321079*n^4+1206165*n^3+
2226053*n^2+2016760*n+716100)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)
/(n+3)^2/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n + 2],  is  the number of syt of [n$4]



32 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1)

            5           4           3            2                         /
    (16766 n  + 169099 n  + 670234 n  + 1302329 n  + 1236768 n + 457380)  /  (9
                                                                         /

                   2
    (n + 2) (n + 3)  (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7)

    (4 + 3 n) (2 n + 7))



                             and in Maple notation



32/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(16766*n^5+169099*n^4+670234*n^3+
1302329*n^2+1236768*n+457380)/(n+2)/(n+3)^2/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(
3*n+7)/(4+3*n)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n + 3],  is  the number of syt of [n$4]



            7            6             5             4              3
32 (231016 n  + 3555230 n  + 23192423 n  + 83045848 n  + 176054487 n

                  2
     + 220620438 n  + 151018290 n + 43451100) (4 n + 1) (4 n + 3) (4 n + 5)

                /                   2
    (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 10)
              /

    (3 n + 7) (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



32/9*(231016*n^7+3555230*n^6+23192423*n^5+83045848*n^4+176054487*n^3+220620438*
n^2+151018290*n+43451100)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+11
)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n + 4],  is  the number of syt of [n$4]



            7            6             5              4              3
32 (327904 n  + 5120560 n  + 33842742 n  + 122594717 n  + 262565798 n

                  2
     + 331979847 n  + 229001580 n + 66320100) (4 n + 1) (4 n + 3) (4 n + 5)

                /                   2
    (4 n + 7)  /  (9 (n + 2) (n + 3)  (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 10)
              /

    (3 n + 7) (4 + 3 n) (2 n + 7) (5 + 2 n))



                             and in Maple notation



32/9*(327904*n^7+5120560*n^6+33842742*n^5+122594717*n^4+262565798*n^3+331979847
*n^2+229001580*n+66320100)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(3*n+
11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n],  is  the number of syt of [n$4]



                                                       2
         2 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (143 n  + 481 n + 336)
         --------------------------------------------------------------
             (n + 4) (n + 3) (n + 2) (2 n + 7) (5 + 2 n) (2 n + 3)



                             and in Maple notation



2*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(143*n^2+481*n+336)/(n+4)/(n+3)/(n+2)/(2*n+7)/(
5+2*n)/(2*n+3)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, n],  is  the number of syt of [n$4]



2 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1)

             5            4            3            2
    (172109 n  + 1555944 n  + 5475947 n  + 9379056 n  + 7820624 n + 2540160)/(9

    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 4) (n + 3) (n + 2) (2 n + 7)

    (5 + 2 n))



                             and in Maple notation



2/9*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(172109*n^5+1555944*n^4+5475947*n^3+9379056*n
^2+7820624*n+2540160)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+4)/(n+3)/(n+2)/(2*n+7)
/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, 1 + n],  is  the number of syt of [n$4]



          5          4           3           2
20 (7679 n  + 71634 n  + 261677 n  + 468186 n  + 410504 n + 141120) (4 n + 1)

    (4 n + 3) (4 n + 5) (4 n + 7)/(9 (n + 4) (n + 3) (n + 2) (3 n + 7)

    (4 + 3 n) (3 n + 8) (3 n + 5) (2 n + 7) (5 + 2 n))



                             and in Maple notation



20/9*(7679*n^5+71634*n^4+261677*n^3+468186*n^2+410504*n+141120)*(4*n+1)*(4*n+3)
*(4*n+5)*(4*n+7)/(n+4)/(n+3)/(n+2)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(2*n+7)/(5+2
*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, n],  is  the number of syt of [n$4]



                                                          6            5
8 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (81383 n  + 1008568 n

                4             3             2                           /
     + 5071039 n  + 13238042 n  + 18925128 n  + 14051520 n + 4233600)  /  (9
                                                                      /

                                                                      2
    (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 2) (n + 3)  (n + 4)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



8/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(81383*n^6+1008568*n^5+5071039*n^4+
13238042*n^3+18925128*n^2+14051520*n+4233600)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/
(3*n+5)/(n+2)/(n+3)^2/(n+4)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, 1 + n],  is  the number of syt of [n$4]



                                                   6           5            4
40 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (32947 n  + 418275 n  + 2163649 n

                3            2                          /
     + 5837697 n  + 8665864 n  + 6710688 n + 2116800)  /  (9 (3 n + 10)
                                                      /

                                                   2
    (3 n + 7) (4 + 3 n) (3 n + 8) (3 n + 5) (n + 3)  (n + 4) (2 n + 7)

    (5 + 2 n))



                             and in Maple notation



40/9*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(32947*n^6+418275*n^5+2163649*n^4+5837697*
n^3+8665864*n^2+6710688*n+2116800)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+8)/(3*n+5)/(n+
3)^2/(n+4)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, n + 2],  is  the number of syt of [n$4]



                                                             6           5
16 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (34549 n  + 455029 n

                4            3             2                          /
     + 2449117 n  + 6889651 n  + 10670734 n  + 8611960 n + 2822400)  /  (9
                                                                    /

                   2
    (n + 2) (n + 3)  (n + 4) (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (4 + 3 n)

    (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(34549*n^6+455029*n^5+2449117*n^4+
6889651*n^3+10670734*n^2+8611960*n+2822400)/(n+2)/(n+3)^2/(n+4)/(3*n+8)/(3*n+5)
/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, n],  is  the number of syt of [n$4]



                                                           6            5
40 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (83935 n  + 1047117 n

                4             3             2                           /
     + 5307389 n  + 13989891 n  + 20231364 n  + 15223824 n + 4656960)  /  (9
                                                                      /

                                                                         2
    (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)

    (n + 4) (2 n + 7) (5 + 2 n))



                             and in Maple notation



40/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(83935*n^6+1047117*n^5+5307389*n^4+
13989891*n^3+20231364*n^2+15223824*n+4656960)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+11)
/(3*n+8)/(3*n+5)/(n+3)^2/(n+4)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, 1 + n],  is  the number of syt of [n$4]



                                                             6           5
80 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (22847 n  + 292131 n

                4            3            2                          /
     + 1523515 n  + 4148405 n  + 6220278 n  + 4868344 n + 1552320)  /  (9
                                                                   /

                                                                         2
    (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)

    (n + 4) (2 n + 7) (5 + 2 n))



                             and in Maple notation



80/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(22847*n^6+292131*n^5+1523515*n^4+
4148405*n^3+6220278*n^2+4868344*n+1552320)/(3*n+10)/(3*n+7)/(4+3*n)/(3*n+11)/(3
*n+8)/(3*n+5)/(n+3)^2/(n+4)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, n + 2],  is  the number of syt of [n$4]



                                                              6            5
16 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (108941 n  + 1444773 n

                4             3             2                           /
     + 7833845 n  + 22208595 n  + 34670414 n  + 28203192 n + 9313920)  /  (9
                                                                      /

                                     2
    (n + 2) (2 n + 7) (n + 4) (n + 3)  (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n)

    (3 n + 7) (3 n + 10))



                             and in Maple notation



16/9*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(108941*n^6+1444773*n^5+7833845*n^
4+22208595*n^3+34670414*n^2+28203192*n+9313920)/(n+2)/(2*n+7)/(n+4)/(n+3)^2/(3*
n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(3*n+7)/(3*n+10)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, n + 3],  is  the number of syt of [n$4]



            7            6             5             4              3
64 (168248 n  + 2686492 n  + 18123759 n  + 66901710 n  + 145772677 n

                  2
     + 187214918 n  + 130981956 n + 38419920) (4 n + 1) (4 n + 3) (4 n + 5)

                /                   2
    (4 n + 7)  /  (9 (n + 2) (n + 3)  (n + 4) (3 n + 10) (3 n + 7) (4 + 3 n)
              /

    (3 n + 11) (3 n + 8) (3 n + 5) (2 n + 7))



                             and in Maple notation



64/9*(168248*n^7+2686492*n^6+18123759*n^5+66901710*n^4+145772677*n^3+187214918*
n^2+130981956*n+38419920)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+2)/(n+3)^2/(n+4)/(
3*n+10)/(3*n+7)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n],  is  the number of syt of [n$4]



           6           5            4            3             2
80 (52685 n  + 661917 n  + 3382319 n  + 8999331 n  + 13154084 n  + 10017504 n

                                                                            /
     + 3104640) (4 n + 1) (4 n + 3) (1 + n) (4 n + 5) (4 n + 7) (4 n + 9)  /  (
                                                                          /

                                                                            2
    27 (3 n + 10) (3 n + 7) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 4)

           2
    (n + 3)  (2 n + 7) (5 + 2 n))



                             and in Maple notation



80/27*(52685*n^6+661917*n^5+3382319*n^4+8999331*n^3+13154084*n^2+10017504*n+
3104640)*(4*n+1)*(4*n+3)*(1+n)*(4*n+5)*(4*n+7)*(4*n+9)/(3*n+10)/(3*n+7)/(4+3*n)
/(3*n+11)/(3*n+8)/(3*n+5)/(n+4)^2/(n+3)^2/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, 1 + n],  is  the number of syt of [n$4]



                                                             6           5
80 (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (60613 n  + 781047 n

                4             3             2                           /
     + 4107559 n  + 11285625 n  + 17083492 n  + 13501248 n + 4346496)  /  (27
                                                                      /

                     2        2
    (2 n + 7) (n + 3)  (n + 4)  (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n)

    (3 n + 7) (3 n + 10))



                             and in Maple notation



80/27*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(60613*n^6+781047*n^5+4107559*n^4
+11285625*n^3+17083492*n^2+13501248*n+4346496)/(2*n+7)/(n+3)^2/(n+4)^2/(3*n+5)/
(3*n+8)/(3*n+11)/(4+3*n)/(3*n+7)/(3*n+10)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n + 2],  is  the number of syt of [n$4]



                                                                        6
64 (4 n + 11) (4 n + 9) (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (39368 n

               5            4            3             2
     + 526299 n  + 2876870 n  + 8222085 n  + 12938522 n  + 10606296 n + 3528000

        /                    2        2
    )  /  (27 (n + 2) (n + 4)  (n + 3)  (3 n + 11) (3 n + 8) (3 n + 5)
      /

    (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 7))



                             and in Maple notation



64/27*(4*n+11)*(4*n+9)*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(39368*n^6+526299*n^5+
2876870*n^4+8222085*n^3+12938522*n^2+10606296*n+3528000)/(n+2)/(n+4)^2/(n+3)^2/
(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n + 3],  is  the number of syt of [n$4]



             7             6              5              4               3
64 (1134632 n  + 18244780 n  + 123906113 n  + 460278460 n  + 1008896453 n

                   2
     + 1302996430 n  + 916400952 n + 270103680) (4 n + 1) (4 n + 3) (4 n + 5)

                /                            2
    (4 n + 7)  /  (27 (n + 3) (n + 2) (n + 4)  (3 n + 11) (3 n + 8) (3 n + 5)
              /

    (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 7))



                             and in Maple notation



64/27*(1134632*n^7+18244780*n^6+123906113*n^5+460278460*n^4+1008896453*n^3+
1302996430*n^2+916400952*n+270103680)*(4*n+1)*(4*n+3)*(4*n+5)*(4*n+7)/(n+3)/(n+
2)/(n+4)^2/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n + 4],  is  the number of syt of [n$4]



             8             7              6               5               4
64 (1639376 n  + 31727128 n  + 265249634 n  + 1250187037 n  + 3629870384 n

                   3               2
     + 6640602667 n  + 7464915126 n  + 4706349768 n + 1271350080) (4 n + 1)

                                    /                    2        2
    (4 n + 3) (4 n + 5) (4 n + 7)  /  (27 (n + 2) (n + 4)  (n + 3)  (3 n + 11)
                                  /

    (3 n + 8) (3 n + 5) (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 7))



                             and in Maple notation



64/27*(1639376*n^8+31727128*n^7+265249634*n^6+1250187037*n^5+3629870384*n^4+
6640602667*n^3+7464915126*n^2+4706349768*n+1271350080)*(4*n+1)*(4*n+3)*(4*n+5)*
(4*n+7)/(n+2)/(n+4)^2/(n+3)^2/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+10)/(3*n+7)/(4+3*n)
/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n],  is  the number of syt of [n$4]



                                                        2
          70 (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (14 n  + 49 n + 36)
          ------------------------------------------------------------
             (2 n + 9) (2 n + 7) (5 + 2 n) (n + 2) (n + 3) (n + 4)



                             and in Maple notation



70*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(14*n^2+49*n+36)/(2*n+9)/(2*n+7)/(5+2*n)/(n+2)
/(n+3)/(n+4)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2, n],  is  the number of syt of [n$4]



                                                           6            5
20 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (96743 n  + 1205908 n

                4             3             2
     + 6089059 n  + 15935162 n  + 22795248 n  + 16905600 n + 5080320)/(9

    (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 9) (2 n + 7) (5 + 2 n) (3 n + 8)

    (3 n + 5) (n + 2) (n + 3) (n + 4))



                             and in Maple notation



20/9*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(96743*n^6+1205908*n^5+6089059*n^4+
15935162*n^3+22795248*n^2+16905600*n+5080320)/(3*n+10)/(3*n+7)/(4+3*n)/(2*n+9)/
(2*n+7)/(5+2*n)/(3*n+8)/(3*n+5)/(n+2)/(n+3)/(n+4)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2, 1 + n],  is  the number of syt of [n$4]



                                                   6            5            4
40 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (89998 n  + 1147353 n  + 5950504 n

                 3             2
     + 16071087 n  + 23846698 n  + 18439560 n + 5806080)/(9 (n + 4) (n + 3)

    (3 n + 5) (3 n + 8) (5 + 2 n) (2 n + 7) (2 n + 9) (4 + 3 n) (3 n + 7)

    (3 n + 10))



                             and in Maple notation



40/9*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(89998*n^6+1147353*n^5+5950504*n^4+
16071087*n^3+23846698*n^2+18439560*n+5806080)/(n+4)/(n+3)/(3*n+5)/(3*n+8)/(5+2*
n)/(2*n+7)/(2*n+9)/(4+3*n)/(3*n+7)/(3*n+10)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, n],  is  the number of syt of [n$4]



                                                            7            6
16 (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (352258 n  + 5663919 n

                 5              4              3              2
     + 38097160 n  + 138832100 n  + 295829602 n  + 368451661 n  + 248323140 n

                   /
     + 69854400)  /  (3 (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 9) (2 n + 7)
                 /

                                                    2
    (5 + 2 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)  (n + 4))



                             and in Maple notation



16/3*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(352258*n^7+5663919*n^6+38097160*n^5
+138832100*n^4+295829602*n^3+368451661*n^2+248323140*n+69854400)/(3*n+10)/(3*n+
7)/(4+3*n)/(2*n+9)/(2*n+7)/(5+2*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)^2/(n+4)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, 1 + n],  is  the number of syt of [n$4]



                                                              7            6
16 (4 n + 7) (4 n + 5) (4 n + 9) (4 n + 3) (4 n + 1) (185497 n  + 3037941 n

                 5             4              3              2
     + 20877325 n  + 77995875 n  + 171002218 n  + 219944904 n  + 153608400 n

                   /                   2
     + 44906400)  /  (3 (n + 4) (n + 3)  (4 + 3 n) (3 n + 8) (3 n + 11)
                 /

    (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7) (5 + 2 n))



                             and in Maple notation



16/3*(4*n+7)*(4*n+5)*(4*n+9)*(4*n+3)*(4*n+1)*(185497*n^7+3037941*n^6+20877325*n
^5+77995875*n^4+171002218*n^3+219944904*n^2+153608400*n+44906400)/(n+4)/(n+3)^2
/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(2*n+7)/(5+2*n)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, n + 2],  is  the number of syt of [n$4]



                                                             7           6
80 (4 n + 7) (4 n + 5) (4 n + 9) (4 n + 3) (4 n + 1) (32878 n  + 553479 n

                5             4             3             2
     + 3920055 n  + 15128435 n  + 34322467 n  + 45717606 n  + 33048440 n

                  /                           2
     + 9979200)  /  (3 (n + 4) (n + 2) (n + 3)  (4 + 3 n) (3 n + 8) (3 n + 11)
                /

    (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7))



                             and in Maple notation



80/3*(4*n+7)*(4*n+5)*(4*n+9)*(4*n+3)*(4*n+1)*(32878*n^7+553479*n^6+3920055*n^5+
15128435*n^4+34322467*n^3+45717606*n^2+33048440*n+9979200)/(n+4)/(n+2)/(n+3)^2/
(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, n],  is  the number of syt of [n$4]



                                                                      7
80 (4 n + 9) (4 n + 7) (4 n + 5) (1 + n) (4 n + 3) (4 n + 1) (373153 n

                6             5              4              3              2
     + 6119979 n  + 41930959 n  + 155507673 n  + 337078504 n  + 427043844 n

                                 /
     + 292817808 n + 83825280)  /  (27 (3 n + 10) (3 n + 7) (4 + 3 n) (2 n + 9)
                               /

                                                              2        2
    (2 n + 7) (5 + 2 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 4)  (n + 3) )



                             and in Maple notation



80/27*(4*n+9)*(4*n+7)*(4*n+5)*(1+n)*(4*n+3)*(4*n+1)*(373153*n^7+6119979*n^6+
41930959*n^5+155507673*n^4+337078504*n^3+427043844*n^2+292817808*n+83825280)/(3
*n+10)/(3*n+7)/(4+3*n)/(2*n+9)/(2*n+7)/(5+2*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+4)^2
/(n+3)^2




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, 1 + n],  is  the number of syt of [n$4]



                                                               7             6
32 (4 n + 7) (4 n + 5) (4 n + 9) (4 n + 3) (4 n + 1) (1055162 n  + 17630829 n

                  5              4               3               2
     + 123436109 n  + 469269465 n  + 1046064353 n  + 1366978026 n

                                  /            2        2
     + 969241896 n + 287400960)  /  (27 (n + 4)  (n + 3)  (4 + 3 n) (3 n + 8)
                                /

    (3 n + 11) (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7))



                             and in Maple notation



32/27*(4*n+7)*(4*n+5)*(4*n+9)*(4*n+3)*(4*n+1)*(1055162*n^7+17630829*n^6+
123436109*n^5+469269465*n^4+1046064353*n^3+1366978026*n^2+969241896*n+287400960
)/(n+4)^2/(n+3)^2/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(2*
n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, n + 2],  is  the number of syt of [n$4]



                                                                          7
160 (4 n + 7) (4 n + 5) (4 n + 11) (4 n + 9) (4 n + 3) (4 n + 1) (106087 n

                6             5             4              3              2
     + 1821549 n  + 13137649 n  + 51558615 n  + 118802608 n  + 160530036 n

                                 /            2                2
     + 117577296 n + 35925120)  /  (27 (n + 4)  (n + 2) (n + 3)  (4 + 3 n)
                               /

    (3 n + 8) (3 n + 11) (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7))



                             and in Maple notation



160/27*(4*n+7)*(4*n+5)*(4*n+11)*(4*n+9)*(4*n+3)*(4*n+1)*(106087*n^7+1821549*n^6
+13137649*n^5+51558615*n^4+118802608*n^3+160530036*n^2+117577296*n+35925120)/(n
+4)^2/(n+2)/(n+3)^2/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(
2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, n + 3],  is  the number of syt of [n$4]



                                                      8             7
320 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (1417664 n  + 28134352 n

                  6               5               4               3
     + 240566474 n  + 1156912099 n  + 3420030482 n  + 6357927403 n

                   2                                /            2
     + 7249955946 n  + 4629115656 n + 1264564224)  /  (27 (n + 4)  (n + 2)
                                                  /

    (n + 3) (4 + 3 n) (3 n + 8) (3 n + 11) (3 n + 5) (3 n + 7) (3 n + 10)

    (2 n + 9) (2 n + 7))



                             and in Maple notation



320/27*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(1417664*n^8+28134352*n^7+240566474*n^6+
1156912099*n^5+3420030482*n^4+6357927403*n^3+7249955946*n^2+4629115656*n+
1264564224)/(n+4)^2/(n+2)/(n+3)/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+
10)/(2*n+9)/(2*n+7)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n],  is  the number of syt of [n$4]



             7             6              5              4               3
64 (1152687 n  + 18991409 n  + 130815964 n  + 488183470 n  + 1065845273 n

                   2
     + 1361478201 n  + 942177636 n + 272432160) (4 n + 1) (4 n + 3) (1 + n)

                                    /
    (4 n + 5) (4 n + 7) (4 n + 9)  /  (27 (3 n + 13) (2 n + 7) (2 n + 9)
                                  /

                                                                         2
    (3 n + 10) (3 n + 7) (3 n + 5) (3 n + 11) (3 n + 8) (4 + 3 n) (n + 4)

           2
    (n + 3) )



                             and in Maple notation



64/27*(1152687*n^7+18991409*n^6+130815964*n^5+488183470*n^4+1065845273*n^3+
1361478201*n^2+942177636*n+272432160)*(4*n+1)*(4*n+3)*(1+n)*(4*n+5)*(4*n+7)*(4*
n+9)/(3*n+13)/(2*n+7)/(2*n+9)/(3*n+10)/(3*n+7)/(3*n+5)/(3*n+11)/(3*n+8)/(4+3*n)
/(n+4)^2/(n+3)^2




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, 1 + n],  is  the number of syt of [n$4]



                                                                         7
160 (4 n + 7) (4 n + 5) (4 n + 11) (4 n + 9) (4 n + 3) (4 n + 1) (91181 n

                6             5             4             3              2
     + 1530882 n  + 10775472 n  + 41208730 n  + 92455219 n  + 121655108 n

                                /           2        2
     + 86877648 n + 25945920)  /  (9 (n + 4)  (n + 3)  (4 + 3 n) (3 n + 8)
                              /

    (3 n + 11) (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7) (3 n + 13))



                             and in Maple notation



160/9*(4*n+7)*(4*n+5)*(4*n+11)*(4*n+9)*(4*n+3)*(4*n+1)*(91181*n^7+1530882*n^6+
10775472*n^5+41208730*n^4+92455219*n^3+121655108*n^2+86877648*n+25945920)/(n+4)
^2/(n+3)^2/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(2*n+7)/(3
*n+13)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n + 2],  is  the number of syt of [n$4]



                                                                          7
320 (4 n + 7) (4 n + 5) (4 n + 11) (4 n + 9) (4 n + 3) (4 n + 1) (295902 n

                6             5              4              3              2
     + 5104049 n  + 36990249 n  + 145901405 n  + 337939713 n  + 459041186 n

                                  /            2
     + 337969416 n + 103783680)  /  (27 (n + 4)  (n + 2) (n + 3) (4 + 3 n)
                                /

    (3 n + 8) (3 n + 11) (3 n + 5) (3 n + 7) (3 n + 10) (2 n + 9) (2 n + 7)

    (3 n + 13))



                             and in Maple notation



320/27*(4*n+7)*(4*n+5)*(4*n+11)*(4*n+9)*(4*n+3)*(4*n+1)*(295902*n^7+5104049*n^6
+36990249*n^5+145901405*n^4+337939713*n^3+459041186*n^2+337969416*n+103783680)/
(n+4)^2/(n+2)/(n+3)/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(
2*n+7)/(3*n+13)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n + 3],  is  the number of syt of [n$4]



                                                      9              8
640 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (4493712 n  + 104084816 n

                   7               6                5                4
     + 1058454100 n  + 6198521855 n  + 23020565596 n  + 56179126793 n

                    3                2                                  /
     + 89994880600 n  + 91139971920 n  + 52866931392 n + 13356959616)  /  (27
                                                                      /

           2
    (n + 4)  (n + 2) (n + 3) (4 + 3 n) (3 n + 8) (3 n + 11) (3 n + 5) (3 n + 7)

    (3 n + 10) (2 n + 9) (2 n + 7) (3 n + 13))



                             and in Maple notation



640/27*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(4493712*n^9+104084816*n^8+1058454100*n^
7+6198521855*n^6+23020565596*n^5+56179126793*n^4+89994880600*n^3+91139971920*n^
2+52866931392*n+13356959616)/(n+4)^2/(n+2)/(n+3)/(4+3*n)/(3*n+8)/(3*n+11)/(3*n+
5)/(3*n+7)/(3*n+10)/(2*n+9)/(2*n+7)/(3*n+13)




Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n + 4],  is  the number of syt of [n$4]



                                                      9              8
64 (4 n + 7) (4 n + 5) (4 n + 3) (4 n + 1) (11538424 n  + 266422180 n

                   7                6                5                 4
     + 2700801350 n  + 15767135395 n  + 58376817827 n  + 142031047855 n

                     3                 2                                   /
     + 226851642775 n  + 229080620730 n  + 132515328024 n + 33392399040)  /  (9
                                                                         /

           2                2
    (n + 4)  (n + 2) (n + 3)  (4 + 3 n) (3 n + 8) (3 n + 11) (3 n + 5)

    (3 n + 7) (3 n + 10) (2 n + 9) (3 n + 13))



                             and in Maple notation



64/9*(4*n+7)*(4*n+5)*(4*n+3)*(4*n+1)*(11538424*n^9+266422180*n^8+2700801350*n^7
+15767135395*n^6+58376817827*n^5+142031047855*n^4+226851642775*n^3+229080620730
*n^2+132515328024*n+33392399040)/(n+4)^2/(n+2)/(n+3)^2/(4+3*n)/(3*n+8)/(3*n+11)
/(3*n+5)/(3*n+7)/(3*n+10)/(2*n+9)/(3*n+13)




                           -------------------------



         This ends this article that took, 48.184, seconds to produce.