Conjectured Explicit Formulas Enumerating 6-rowed Truncated Standard Young \
    Tableau of shape [n,n+a1,n+a2,n+a3,n+a4,n+a5] for a from 0<=a1,a2,a3,a4,\

    a5<= , 2, and a5<a4 divided by  the number of Standard Young Tableaux of shape [n, n, n, n, n, n]



                              By Shalosh B. Ekhad





Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n, 1 + n, n],  is



              72 (6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5)
              ----------------------------------------------------
              5 (5 n + 2) (5 n + 4) (5 n + 3) (5 n + 1) (6 + 5 n)



                             and in Maple notation



72/5*(6*n+1)*(3*n+1)*(2*n+1)*(3*n+2)*(6*n+5)/(5*n+2)/(5*n+4)/(5*n+3)/(5*n+1)/(6
+5*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n, n + 2, n],  is



        72 (11 n + 7) (6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5)
        ---------------------------------------------------------------
         5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 7) (5 n + 1) (6 + 5 n)



                             and in Maple notation



72/5*(11*n+7)*(6*n+1)*(3*n+1)*(2*n+1)*(3*n+2)*(6*n+5)/(5*n+4)/(5*n+3)/(5*n+2)/(
5*n+7)/(5*n+1)/(6+5*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n, n + 2, 1 + n],  is



        144 (8 n + 7) (6 n + 1) (3 n + 1) (2 n + 1) (3 n + 2) (6 n + 5)
        ---------------------------------------------------------------
         5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 7) (5 n + 1) (6 + 5 n)



                             and in Maple notation



144/5*(8*n+7)*(6*n+1)*(3*n+1)*(2*n+1)*(3*n+2)*(6*n+5)/(5*n+4)/(5*n+3)/(5*n+2)/(
5*n+7)/(5*n+1)/(6+5*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, 1 + n, 1 + n, n],  is



                                 4          3          2
    3 (3 n + 1) (6 n + 1) (9763 n  + 47673 n  + 82520 n  + 59424 n + 15120)
    -----------------------------------------------------------------------
        16 (9 + 4 n) (5 + 4 n) (4 n + 1) (7 + 4 n) (4 n + 3) (2 n + 3)



                             and in Maple notation



3/16*(3*n+1)*(6*n+1)*(9763*n^4+47673*n^3+82520*n^2+59424*n+15120)/(9+4*n)/(5+4*
n)/(4*n+1)/(7+4*n)/(4*n+3)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, 1 + n, n + 2, n],  is



                                   4          3          2
     21 (3 n + 1) (6 n + 1) (2879 n  + 11619 n  + 16390 n  + 9852 n + 2160)
     ----------------------------------------------------------------------
         16 (9 + 4 n) (5 + 4 n) (4 n + 1) (7 + 4 n) (4 n + 3) (2 n + 3)



                             and in Maple notation



21/16*(3*n+1)*(6*n+1)*(2879*n^4+11619*n^3+16390*n^2+9852*n+2160)/(9+4*n)/(5+4*n
)/(4*n+1)/(7+4*n)/(4*n+3)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, 1 + n, n + 2, 1 + n],  is



                                                        2
        9 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (277 n  + 779 n + 504)
        ----------------------------------------------------------------
         8 (9 + 4 n) (5 + 4 n) (4 n + 1) (7 + 4 n) (4 n + 3) (2 n + 3)



                             and in Maple notation



9/8*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(277*n^2+779*n+504)/(9+4*n)/(5+4*n)/(4*n+1)
/(7+4*n)/(4*n+3)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 2, 1 + n, n],  is



15 (3 n + 1) (6 n + 1)

            5           4           3           2
    (22643 n  + 134321 n  + 300343 n  + 315739 n  + 157674 n + 30240)/(16

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



                             and in Maple notation



15/16*(3*n+1)*(6*n+1)*(22643*n^5+134321*n^4+300343*n^3+315739*n^2+157674*n+
30240)/(7+4*n)/(4*n+3)/(9+4*n)/(5+4*n)/(4*n+1)/(2*n+5)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 2, n + 2, n],  is



63 (3 n + 2) (3 n + 1) (6 n + 1)

           4          3          2
    (3473 n  + 15066 n  + 22936 n  + 14985 n + 3600)/(16 (7 + 4 n) (4 n + 3)

    (9 + 4 n) (5 + 4 n) (4 n + 1) (2 n + 5) (2 n + 3))



                             and in Maple notation



63/16*(3*n+2)*(3*n+1)*(6*n+1)*(3473*n^4+15066*n^3+22936*n^2+14985*n+3600)/(7+4*
n)/(4*n+3)/(9+4*n)/(5+4*n)/(4*n+1)/(2*n+5)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 2, n + 2, 1 + n],  is



                                                          2
       945 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (1 + n)  (31 n + 60)
    -----------------------------------------------------------------------
    8 (7 + 4 n) (4 n + 3) (9 + 4 n) (5 + 4 n) (4 n + 1) (2 n + 5) (2 n + 3)



                             and in Maple notation



945/8*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(1+n)^2*(31*n+60)/(7+4*n)/(4*n+3)/(9+4*n)
/(5+4*n)/(4*n+1)/(2*n+5)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n, 1 + n, n],  is



                                 9              8               7
8 (2 n + 1) (6 n + 1) (22443559 n  + 274421844 n  + 1419960640 n

                   6               5               4               3
     + 4066786654 n  + 7078488837 n  + 7732394390 n  + 5275185316 n

                   2
     + 2154381960 n  + 474112800 n + 42336000)/(15 (10 + 3 n) (3 n + 7)

    (3 n + 4) (6 + 5 n) (5 n + 1) (3 n + 8) (3 n + 5) (5 n + 7) (5 n + 2)

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



                             and in Maple notation



8/15*(2*n+1)*(6*n+1)*(22443559*n^9+274421844*n^8+1419960640*n^7+4066786654*n^6+
7078488837*n^5+7732394390*n^4+5275185316*n^3+2154381960*n^2+474112800*n+
42336000)/(10+3*n)/(3*n+7)/(3*n+4)/(6+5*n)/(5*n+1)/(3*n+8)/(3*n+5)/(5*n+7)/(5*n
+2)/(5*n+3)/(5*n+4)








Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, 1 + n, 1 + n, n],  is



                                         3        2
            56 (2 n + 1) (6 n + 1) (151 n  + 809 n  + 1300 n + 600)
            -------------------------------------------------------
             3 (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 8) (3 n + 5)



                             and in Maple notation



56/3*(2*n+1)*(6*n+1)*(151*n^3+809*n^2+1300*n+600)/(10+3*n)/(3*n+7)/(3*n+4)/(3*n
+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, 1 + n, n + 2, n],  is



                                7            6             5             4
56 (2 n + 1) (6 n + 1) (951349 n  + 9164397 n  + 36412183 n  + 77697687 n

                 3             2
     + 96508204 n  + 69991380 n  + 27518400 n + 4536000)/(15 (10 + 3 n)

    (3 n + 7) (3 n + 4) (6 + 5 n) (5 n + 9) (5 n + 4) (3 n + 8) (3 n + 5)

    (5 n + 7))



                             and in Maple notation



56/15*(2*n+1)*(6*n+1)*(951349*n^7+9164397*n^6+36412183*n^5+77697687*n^4+
96508204*n^3+69991380*n^2+27518400*n+4536000)/(10+3*n)/(3*n+7)/(3*n+4)/(6+5*n)/
(5*n+9)/(5*n+4)/(3*n+8)/(3*n+5)/(5*n+7)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, 1 + n, n + 2, 1 + n],  is



                     5          4           3           2
112 (6 n + 7) (6589 n  + 52314 n  + 158795 n  + 231102 n  + 161400 n + 43200)

    (6 n + 1) (2 n + 1) (6 n + 5)/(5 (10 + 3 n) (3 n + 7) (3 n + 4) (6 + 5 n)

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



                             and in Maple notation



112/5*(6*n+7)*(6589*n^5+52314*n^4+158795*n^3+231102*n^2+161400*n+43200)*(6*n+1)
*(2*n+1)*(6*n+5)/(10+3*n)/(3*n+7)/(3*n+4)/(6+5*n)/(5*n+9)/(5*n+4)/(3*n+8)/(3*n+
5)/(5*n+7)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 2, 1 + n, n],  is



                                        3         2
           56 (2 n + 1) (6 n + 1) (821 n  + 3109 n  + 3450 n + 1200)
           ---------------------------------------------------------
             3 (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 8) (3 n + 5)



                             and in Maple notation



56/3*(2*n+1)*(6*n+1)*(821*n^3+3109*n^2+3450*n+1200)/(10+3*n)/(3*n+7)/(3*n+4)/(3
*n+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 2, n + 2, n],  is



                                             3         2
      16 (6 n + 5) (2 n + 1) (6 n + 1) (909 n  + 3599 n  + 4342 n + 1680)
      -------------------------------------------------------------------
         3 (n + 2) (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 8) (3 n + 5)



                             and in Maple notation



16/3*(6*n+5)*(2*n+1)*(6*n+1)*(909*n^3+3599*n^2+4342*n+1680)/(n+2)/(10+3*n)/(3*n
+7)/(3*n+4)/(3*n+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 2, n + 2, 1 + n],  is



                2
        32 (41 n  + 134 n + 100) (6 n + 1) (2 n + 1) (6 n + 5) (6 n + 7)
        ----------------------------------------------------------------
           (n + 2) (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 8) (3 n + 5)



                             and in Maple notation



32*(41*n^2+134*n+100)*(6*n+1)*(2*n+1)*(6*n+5)*(6*n+7)/(n+2)/(10+3*n)/(3*n+7)/(3
*n+4)/(3*n+8)/(3*n+5)










Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, 1 + n, 1 + n, n],  is



                                   4         3          2
    280 (2 n + 1) (6 n + 1) (1023 n  + 6792 n  + 15085 n  + 13180 n + 3960)
    -----------------------------------------------------------------------
        3 (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 11) (3 n + 8) (3 n + 5)



                             and in Maple notation



280/3*(2*n+1)*(6*n+1)*(1023*n^4+6792*n^3+15085*n^2+13180*n+3960)/(10+3*n)/(3*n+
7)/(3*n+4)/(3*n+11)/(3*n+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, 1 + n, n + 2, n],  is



                                             9               8               7
16 (6 n + 5) (2 n + 1) (6 n + 1) (105630598 n  + 1458845653 n  + 8730755321 n

                    6                5                4                3
     + 29766898495 n  + 63800019413 n  + 89245506916 n  + 81556902580 n

                    2
     + 46995916944 n  + 15509249280 n + 2235340800)/(15 (6 + 5 n) (5 n + 4)

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

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



                             and in Maple notation



16/15*(6*n+5)*(2*n+1)*(6*n+1)*(105630598*n^9+1458845653*n^8+8730755321*n^7+
29766898495*n^6+63800019413*n^5+89245506916*n^4+81556902580*n^3+46995916944*n^2
+15509249280*n+2235340800)/(6+5*n)/(5*n+4)/(3*n+8)/(3*n+5)/(5*n+7)/(5*n+8)/(5*n
+9)/(10+3*n)/(3*n+11)/(n+2)/(3*n+7)/(3*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, 1 + n, n + 2, 1 + n],  is



                                                     8             7
64 (6 n + 5) (2 n + 1) (6 n + 1) (6 n + 7) (2344914 n  + 30789839 n

                  6              5               4               3
     + 172666631 n  + 540989567 n  + 1036860239 n  + 1245600338 n

                  2
     + 916189592 n  + 377215680 n + 66528000)/(5 (6 + 5 n) (5 n + 4) (3 n + 8)

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

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



                             and in Maple notation



64/5*(6*n+5)*(2*n+1)*(6*n+1)*(6*n+7)*(2344914*n^8+30789839*n^7+172666631*n^6+
540989567*n^5+1036860239*n^4+1245600338*n^3+916189592*n^2+377215680*n+66528000)
/(6+5*n)/(5*n+4)/(3*n+8)/(3*n+5)/(5*n+7)/(5*n+8)/(5*n+9)/(10+3*n)/(3*n+11)/(n+2
)/(3*n+7)/(3*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 2, 1 + n, n],  is



                3         2
     112 (1921 n  + 7656 n  + 8963 n + 3300) (6 n + 1) (2 n + 1) (6 n + 5)
     ---------------------------------------------------------------------
        3 (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 11) (3 n + 8) (3 n + 5)



                             and in Maple notation



112/3*(1921*n^3+7656*n^2+8963*n+3300)*(6*n+1)*(2*n+1)*(6*n+5)/(10+3*n)/(3*n+7)/
(3*n+4)/(3*n+11)/(3*n+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 2, n + 2, n],  is



                                                        2
        560 (1 + n) (6 n + 5) (2 n + 1) (6 n + 1) (237 n  + 453 n + 220)
        ----------------------------------------------------------------
         (10 + 3 n) (3 n + 7) (3 n + 4) (3 n + 11) (3 n + 8) (3 n + 5)



                             and in Maple notation



560*(1+n)*(6*n+5)*(2*n+1)*(6*n+1)*(237*n^2+453*n+220)/(10+3*n)/(3*n+7)/(3*n+4)/
(3*n+11)/(3*n+8)/(3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 2, n + 2, 1 + n],  is



             12320 (6 n + 7) (1 + n) (6 n + 5) (2 n + 1) (6 n + 1)
             -----------------------------------------------------
              (10 + 3 n) (3 n + 7) (3 n + 11) (3 n + 8) (3 n + 5)



                             and in Maple notation



12320*(6*n+7)*(1+n)*(6*n+5)*(2*n+1)*(6*n+1)/(10+3*n)/(3*n+7)/(3*n+11)/(3*n+8)/(
3*n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n, 1 + n, n],  is



                                                   6           5           4
27 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (17919 n  + 184203 n  + 713855 n

                3            2
     + 1316485 n  + 1196026 n  + 492632 n + 70560)/(10 (6 + 5 n) (5 n + 1)

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

    (5 n + 4))



                             and in Maple notation



27/10*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(17919*n^6+184203*n^5+713855*n^4+1316485*
n^3+1196026*n^2+492632*n+70560)/(6+5*n)/(5*n+1)/(5*n+7)/(5*n+2)/(9+2*n)/(2*n+7)
/(2*n+5)/(2*n+3)/(5*n+3)/(5*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n, n + 2, n],  is



                                                    7            6
27 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (208723 n  + 2347925 n

                 5             4             3             2
     + 10375213 n  + 23245463 n  + 28431544 n  + 18954668 n  + 6400464 n

     + 846720)/(10 (5 n + 8) (6 + 5 n) (5 n + 1) (5 n + 7) (5 n + 2) (9 + 2 n)

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



                             and in Maple notation



27/10*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(208723*n^7+2347925*n^6+10375213*n^5+
23245463*n^4+28431544*n^3+18954668*n^2+6400464*n+846720)/(5*n+8)/(6+5*n)/(5*n+1
)/(5*n+7)/(5*n+2)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+3)/(5*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n, n + 2, 1 + n],  is



                                                    7            6            5
27 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (158119 n  + 1841442 n  + 8529792 n

                 4             3             2
     + 20352084 n  + 27008337 n  + 19919202 n  + 7548464 n + 1128960)/(5

    (5 n + 8) (6 + 5 n) (5 n + 1) (5 n + 7) (5 n + 2) (9 + 2 n) (2 n + 7)

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



                             and in Maple notation



27/5*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(158119*n^7+1841442*n^6+8529792*n^5+
20352084*n^4+27008337*n^3+19919202*n^2+7548464*n+1128960)/(5*n+8)/(6+5*n)/(5*n+
1)/(5*n+7)/(5*n+2)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+3)/(5*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, 1 + n, 1 + n, n],  is



                                 10              9               8
3 (3 n + 1) (6 n + 1) (14152448 n   + 231168048 n  + 1625589930 n

                   7                6                5                4
     + 6472027989 n  + 16127277432 n  + 26228200257 n  + 28136760055 n

                    3               2                               /
     + 19626656421 n  + 8510371920 n  + 2070117000 n + 214326000)  /  (16
                                                                  /

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

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



                             and in Maple notation



3/16*(3*n+1)*(6*n+1)*(14152448*n^10+231168048*n^9+1625589930*n^8+6472027989*n^7
+16127277432*n^6+26228200257*n^5+28136760055*n^4+19626656421*n^3+8510371920*n^2
+2070117000*n+214326000)/(9+4*n)/(5+4*n)/(4*n+1)/(9+2*n)/(2*n+7)/(7+4*n)/(4*n+3
)/(2*n+1)/(2*n+5)^2/(2*n+3)^2




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, 1 + n, n + 2, n],  is



                                          9             8              7
9 (3 n + 2) (3 n + 1) (6 n + 1) (3424576 n  + 51138192 n  + 322457502 n

                   6               5               4               3
     + 1125230895 n  + 2391256944 n  + 3206328723 n  + 2712280988 n

                   2                             /
     + 1396725630 n  + 397570950 n + 47628000)  /  (16 (9 + 4 n) (5 + 4 n)
                                               /

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

             2
    (2 n + 3) )



                             and in Maple notation



9/16*(3*n+2)*(3*n+1)*(6*n+1)*(3424576*n^9+51138192*n^8+322457502*n^7+1125230895
*n^6+2391256944*n^5+3206328723*n^4+2712280988*n^3+1396725630*n^2+397570950*n+
47628000)/(9+4*n)/(5+4*n)/(4*n+1)/(9+2*n)/(2*n+7)/(7+4*n)/(4*n+3)/(2*n+1)/(2*n+
5)^2/(2*n+3)^2




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, 1 + n, n + 2, 1 + n],  is



                                                   8           7            6
63 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (62848 n  + 912816 n  + 5536906 n

                 5             4             3             2
     + 18283545 n  + 35851072 n  + 42598764 n  + 29821389 n  + 11177460 n

                  /
     + 1701000)  /  (8 (9 + 4 n) (5 + 4 n) (4 n + 1) (9 + 2 n) (2 n + 7)
                /

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



                             and in Maple notation



63/8*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(62848*n^8+912816*n^7+5536906*n^6+18283545
*n^5+35851072*n^4+42598764*n^3+29821389*n^2+11177460*n+1701000)/(9+4*n)/(5+4*n)
/(4*n+1)/(9+2*n)/(2*n+7)/(7+4*n)/(4*n+3)/(2*n+1)/(2*n+5)^2/(2*n+3)^2




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 2, 1 + n, n],  is



                                            10              9               8
63 (3 n + 2) (3 n + 1) (6 n + 1) (22211264 n   + 378279344 n  + 2783324640 n

                    7                6                5                4
     + 11633129952 n  + 30529801116 n  + 52465521411 n  + 59684226985 n

                    3                2                               /
     + 44319803113 n  + 20543981475 n  + 5365064700 n + 598752000)  /  (64
                                                                   /

    (4 n + 11) (9 + 4 n) (5 + 4 n) (4 n + 1) (9 + 2 n) (2 n + 7) (7 + 4 n)

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



                             and in Maple notation



63/64*(3*n+2)*(3*n+1)*(6*n+1)*(22211264*n^10+378279344*n^9+2783324640*n^8+
11633129952*n^7+30529801116*n^6+52465521411*n^5+59684226985*n^4+44319803113*n^3
+20543981475*n^2+5365064700*n+598752000)/(4*n+11)/(9+4*n)/(5+4*n)/(4*n+1)/(9+2*
n)/(2*n+7)/(7+4*n)/(4*n+3)/(2*n+1)/(2*n+5)^2/(2*n+3)^2








Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n, 1 + n, n],  is



                                                    7           6            5
126 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (90349 n  + 810275 n  + 2899699 n

                4            3            2
     + 5343917 n  + 5434204 n  + 3020192 n  + 843444 n + 90720)/(5 (5 n + 8)

    (6 + 5 n) (5 n + 1) (5 n + 7) (5 n + 2) (9 + 2 n) (2 n + 7) (2 n + 5)

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



                             and in Maple notation



126/5*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(90349*n^7+810275*n^6+2899699*n^5+5343917
*n^4+5434204*n^3+3020192*n^2+843444*n+90720)/(5*n+8)/(6+5*n)/(5*n+1)/(5*n+7)/(5
*n+2)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+3)/(5*n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n, n + 2, n],  is



                                                     8            7
378 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (377834 n  + 3864606 n

                 6             5             4             3             2
     + 16359939 n  + 37416261 n  + 50507421 n  + 41167989 n  + 19779406 n

     + 5117184 n + 544320)/(5 (5 n + 8) (5 n + 3) (6 + 5 n) (5 n + 1) (9 + 2 n)

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



                             and in Maple notation



378/5*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(377834*n^8+3864606*n^7+16359939*n^6+
37416261*n^5+50507421*n^4+41167989*n^3+19779406*n^2+5117184*n+544320)/(5*n+8)/(
5*n+3)/(6+5*n)/(5*n+1)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+9)/(5*n+4)/(5*n+7)/
(5*n+2)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n, n + 2, 1 + n],  is



                                                     8            7
567 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (802197 n  + 8636616 n

                 6             5              4              3             2
     + 38922286 n  + 95955572 n  + 141484393 n  + 127617444 n  + 68624084 n

     + 20042208 n + 2419200)/(10 (5 n + 8) (5 n + 3) (6 + 5 n) (5 n + 1)

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

    (5 n + 2))



                             and in Maple notation



567/10*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(802197*n^8+8636616*n^7+38922286*n^6+
95955572*n^5+141484393*n^4+127617444*n^3+68624084*n^2+20042208*n+2419200)/(5*n+
8)/(5*n+3)/(6+5*n)/(5*n+1)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+9)/(5*n+4)/(5*n
+7)/(5*n+2)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, 1 + n, 1 + n, n],  is



                  189 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1)
                  -------------------------------------------
                   2 (9 + 2 n) (2 n + 7) (2 n + 5) (2 n + 3)



                             and in Maple notation



189/2*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, 1 + n, n + 2, n],  is



189 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1)

           4         3          2
    (1247 n  + 6493 n  + 12552 n  + 10668 n + 3360)/(10 (n + 2) (5 n + 8)

    (6 + 5 n) (9 + 2 n) (2 n + 7) (5 n + 7) (2 n + 5) (2 n + 3))



                             and in Maple notation



189/10*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(1247*n^4+6493*n^3+12552*n^2+10668*n+
3360)/(n+2)/(5*n+8)/(6+5*n)/(9+2*n)/(2*n+7)/(5*n+7)/(2*n+5)/(2*n+3)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, 1 + n, n + 2, 1 + n],  is



378 (13 n + 20) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (6 n + 7) (3 n + 4)/(5

    (2 n + 5) (5 n + 7) (2 n + 7) (9 + 2 n) (6 + 5 n) (5 n + 8) (n + 2))



                             and in Maple notation



378/5*(13*n+20)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(6*n+7)*(3*n+4)/(2*n+5)/(5*n+7)
/(2*n+7)/(9+2*n)/(6+5*n)/(5*n+8)/(n+2)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 2, 1 + n, n],  is



                                                   7            6             5
9 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (456166 n  + 5728953 n  + 30362425 n

                 4              3              2                            /
     + 87938385 n  + 150174109 n  + 151062462 n  + 82793340 n + 19051200)  /  (
                                                                          /

                2                    2
    32 (2 n + 3)  (2 n + 7) (2 n + 5)  (n + 2) (n + 3) (9 + 4 n) (7 + 4 n)

    (5 + 4 n) (9 + 2 n))



                             and in Maple notation



9/32*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(456166*n^7+5728953*n^6+30362425*n^5+
87938385*n^4+150174109*n^3+151062462*n^2+82793340*n+19051200)/(2*n+3)^2/(2*n+7)
/(2*n+5)^2/(n+2)/(n+3)/(9+4*n)/(7+4*n)/(5+4*n)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 2, n + 2, n],  is



            6            5            4             3             2
63 (122654 n  + 1288363 n  + 5540632 n  + 12485933 n  + 15554958 n

     + 10162260 n + 2721600) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (1 + n)

       /              2                    2
      /  (32 (2 n + 3)  (2 n + 7) (2 n + 5)  (n + 2) (n + 3) (9 + 4 n)
     /

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



                             and in Maple notation



63/32*(122654*n^6+1288363*n^5+5540632*n^4+12485933*n^3+15554958*n^2+10162260*n+
2721600)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(1+n)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2
)/(n+3)/(9+4*n)/(7+4*n)/(5+4*n)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 2, n + 2, 1 + n],  is



           5          4          3           2
135 (2594 n  + 24685 n  + 92692 n  + 171683 n  + 156942 n + 56700) (6 n + 7)

                                                        /              2
    (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (3 n + 4)  /  (16 (2 n + 3)
                                                      /

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

    (9 + 2 n))



                             and in Maple notation



135/16*(2594*n^5+24685*n^4+92692*n^3+171683*n^2+156942*n+56700)*(6*n+7)*(6*n+1)
*(3*n+1)*(3*n+2)*(6*n+5)*(3*n+4)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2)/(n+3)/(9+4*n
)/(7+4*n)/(5+4*n)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n, 1 + n, n],  is



                                                      9              8
27 (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (27768659 n  + 328813242 n

                   7               6               5               4
     + 1643517598 n  + 4545876352 n  + 7648422895 n  + 8089157158 n

                   3               2
     + 5355680192 n  + 2130721408 n  + 459639936 n + 40642560)/(10 (6 + 5 n)

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

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



                             and in Maple notation



27/10*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(27768659*n^9+328813242*n^8+1643517598*n^
7+4545876352*n^6+7648422895*n^5+8089157158*n^4+5355680192*n^3+2130721408*n^2+
459639936*n+40642560)/(6+5*n)/(5*n+2)/(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+9)/(
5*n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n, n + 2, n],  is



               10               9                8                7
27 (371347461 n   + 4982139798 n  + 28860533302 n  + 95058298608 n

                     6                 5                 4                 3
     + 197061918585 n  + 268488085362 n  + 243267536748 n  + 144580389192 n

                    2
     + 53854356704 n  + 11323146240 n + 1016064000) (6 n + 5) (3 n + 2)

    (3 n + 1) (6 n + 1)/(50 (6 + 5 n) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 7)

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

    (9 + 2 n))



                             and in Maple notation



27/50*(371347461*n^10+4982139798*n^9+28860533302*n^8+95058298608*n^7+
197061918585*n^6+268488085362*n^5+243267536748*n^4+144580389192*n^3+53854356704
*n^2+11323146240*n+1016064000)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)/(6+5*n)/(5*n+2)/
(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+9)/(5*n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+2)/(
n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n, n + 2, 1 + n],  is



               10               9                8                7
54 (154462354 n   + 2179046567 n  + 13379160618 n  + 47106778362 n

                     6                 5                 4                 3
     + 105328144530 n  + 156198968583 n  + 155433634562 n  + 102313414408 n

                    2
     + 42526048896 n  + 10039317120 n + 1016064000) (6 n + 5) (3 n + 2)

    (3 n + 1) (6 n + 1)/(25 (6 + 5 n) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 7)

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

    (9 + 2 n))



                             and in Maple notation



54/25*(154462354*n^10+2179046567*n^9+13379160618*n^8+47106778362*n^7+
105328144530*n^6+156198968583*n^5+155433634562*n^4+102313414408*n^3+42526048896
*n^2+10039317120*n+1016064000)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)/(6+5*n)/(5*n+2)/
(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+9)/(5*n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+2)/(
n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, 1 + n, 1 + n, n],  is



                                                3         2
9 (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (223 n  + 1437 n  + 2768 n + 1512)/(2

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



                             and in Maple notation



9/2*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(223*n^3+1437*n^2+2768*n+1512)/(2*n+3)/(2*n
+7)/(2*n+5)/(n+2)/(n+3)/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, 1 + n, n + 2, n],  is



                                                            7            6
63 (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (563074 n  + 7213461 n

                 5              4              3              2
     + 38563261 n  + 111821733 n  + 190364233 n  + 190608942 n  + 104096016 n

     + 23950080)/(5 (5 n + 8) (5 n + 11) (6 + 5 n) (9 + 2 n) (2 n + 7)

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



                             and in Maple notation



63/5*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(563074*n^7+7213461*n^6+38563261*n^5
+111821733*n^4+190364233*n^3+190608942*n^2+104096016*n+23950080)/(5*n+8)/(5*n+
11)/(6+5*n)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+9)/(5*n+7)/(n+2)/(n+3)/(n+4)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, 1 + n, n + 2, 1 + n],  is



           5          4           3           2
216 (7337 n  + 75857 n  + 303907 n  + 591595 n  + 561084 n + 207900) (6 n + 7)

    (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (3 n + 4)/(5 (6 + 5 n) (5 n + 7)

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

    (9 + 2 n))



                             and in Maple notation



216/5*(7337*n^5+75857*n^4+303907*n^3+591595*n^2+561084*n+207900)*(6*n+7)*(6*n+1
)*(3*n+1)*(3*n+2)*(6*n+5)*(3*n+4)/(6+5*n)/(5*n+7)/(5*n+8)/(5*n+9)/(5*n+11)/(2*n
+7)/(2*n+5)/(n+2)/(n+3)/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 2, 1 + n, n],  is



                                                             2
    315 (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (14 n  + 49 n + 36)
    -----------------------------------------------------------------------
        (2 n + 3) (2 n + 7) (2 n + 5) (n + 2) (n + 3) (n + 4) (9 + 2 n)



                             and in Maple notation



315*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(14*n^2+49*n+36)/(2*n+3)/(2*n+7)/(2*n
+5)/(n+2)/(n+3)/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 2, n + 2, n],  is



    270 (6 n + 7) (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (6 + 5 n)
    -----------------------------------------------------------------------
        (2 n + 3) (2 n + 7) (2 n + 5) (n + 2) (n + 3) (n + 4) (9 + 2 n)



                             and in Maple notation



270*(6*n+7)*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(6+5*n)/(2*n+3)/(2*n+7)/(2*n+
5)/(n+2)/(n+3)/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 2, n + 2, 1 + n],  is



        378 (6 n + 7) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (3 n + 4)
        ---------------------------------------------------------------
             (2 n + 7) (2 n + 5) (n + 2) (n + 3) (n + 4) (9 + 2 n)



                             and in Maple notation



378*(6*n+7)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(3*n+4)/(2*n+7)/(2*n+5)/(n+2)/(n+3)
/(n+4)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n, 1 + n, n],  is



                                                              7            6
81 (6 n + 5) (3 n + 2) (2 n + 1) (3 n + 1) (6 n + 1) (182279 n  + 2250718 n

                 5             4             3             2
     + 11054570 n  + 27868772 n  + 38641903 n  + 29025918 n  + 10645600 n

     + 1411200)/(10 (6 + 5 n) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 7) (5 n + 8)

    (5 n + 1) (2 n + 3) (2 n + 7) (2 n + 5) (n + 5) (9 + 2 n))



                             and in Maple notation



81/10*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(6*n+1)*(182279*n^7+2250718*n^6+11054570*
n^5+27868772*n^4+38641903*n^3+29025918*n^2+10645600*n+1411200)/(6+5*n)/(5*n+2)/
(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n, n + 2, n],  is



             9             8              7               6               5
81 (4436659 n  + 62826826 n  + 370664169 n  + 1195839846 n  + 2324650857 n

                   4               3              2
     + 2822842938 n  + 2141066507 n  + 978545558 n  + 244525680 n + 25401600)

    (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5)/(10 (6 + 5 n) (5 n + 2) (5 n + 3)

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

    (2 n + 5) (n + 5) (9 + 2 n))



                             and in Maple notation



81/10*(4436659*n^9+62826826*n^8+370664169*n^7+1195839846*n^6+2324650857*n^5+
2822842938*n^4+2141066507*n^3+978545558*n^2+244525680*n+25401600)*(6*n+1)*(3*n+
1)*(3*n+2)*(6*n+5)/(6+5*n)/(5*n+2)/(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+9)/(5*n
+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n, n + 2, 1 + n],  is



             9             8              7               6               5
63 (4463324 n  + 65697767 n  + 406831522 n  + 1393094624 n  + 2910295402 n

                   4               3               2
     + 3848635277 n  + 3221174744 n  + 1643543340 n  + 462628800 n + 54432000)

    (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5)/(5 (6 + 5 n) (5 n + 2) (5 n + 3)

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

    (2 n + 5) (n + 5) (9 + 2 n))



                             and in Maple notation



63/5*(4463324*n^9+65697767*n^8+406831522*n^7+1393094624*n^6+2910295402*n^5+
3848635277*n^4+3221174744*n^3+1643543340*n^2+462628800*n+54432000)*(6*n+1)*(3*n
+1)*(3*n+2)*(6*n+5)/(6+5*n)/(5*n+2)/(5*n+3)/(5*n+4)/(5*n+7)/(5*n+8)/(5*n+9)/(5*
n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, 1 + n, 1 + n, n],  is



              11              10               9               8
315 (9299264 n   + 181182160 n   + 1546104240 n  + 7614638688 n

                    7                6                5                4
     + 23999146092 n  + 50701613481 n  + 73073045248 n  + 71650129586 n

                    3                2
     + 46716549036 n  + 19241226405 n  + 4493377800 n + 449064000) (6 n + 1)

                          /
    (3 n + 1) (3 n + 2)  /  (64 (2 n + 1) (5 + 4 n) (7 + 4 n) (9 + 4 n)
                        /

                                  2                    2
    (4 n + 11) (4 n + 1) (2 n + 3)  (2 n + 7) (2 n + 5)  (n + 5) (4 n + 3)

    (9 + 2 n))



                             and in Maple notation



315/64*(9299264*n^11+181182160*n^10+1546104240*n^9+7614638688*n^8+23999146092*n
^7+50701613481*n^6+73073045248*n^5+71650129586*n^4+46716549036*n^3+19241226405*
n^2+4493377800*n+449064000)*(6*n+1)*(3*n+1)*(3*n+2)/(2*n+1)/(5+4*n)/(7+4*n)/(9+
4*n)/(4*n+11)/(4*n+1)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+5)/(4*n+3)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, 1 + n, n + 2, n],  is



              11              10               9                8
45 (24653408 n   + 494080136 n   + 4351898204 n  + 22207203906 n

                    7                 6                 5                 4
     + 72821363061 n  + 160808642256 n  + 243495326776 n  + 252256344244 n

                     3                2
     + 174836620231 n  + 77035197378 n  + 19366603200 n + 2095632000) (6 n + 1)

                                    /
    (3 n + 1) (3 n + 2) (6 n + 5)  /  (64 (2 n + 1) (5 + 4 n) (7 + 4 n)
                                  /

                                            2                    2
    (9 + 4 n) (4 n + 11) (4 n + 1) (2 n + 3)  (2 n + 7) (2 n + 5)  (n + 2)

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



                             and in Maple notation



45/64*(24653408*n^11+494080136*n^10+4351898204*n^9+22207203906*n^8+72821363061*
n^7+160808642256*n^6+243495326776*n^5+252256344244*n^4+174836620231*n^3+
77035197378*n^2+19366603200*n+2095632000)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)/(2*n+
1)/(5+4*n)/(7+4*n)/(9+4*n)/(4*n+11)/(4*n+1)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2)/(
n+5)/(4*n+3)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, 1 + n, n + 2, 1 + n],  is



             11               10                9                8
9 (97984576 n   + 2020305808 n   + 18392244112 n  + 97512686328 n

                     7                 6                  5                  4
     + 334201671840 n  + 776428755687 n  + 1245716742443 n  + 1377552520517 n

                      3                 2
     + 1026405069389 n  + 489144217500 n  + 133561801800 n + 15717240000)

                                              /
    (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5)  /  (32 (2 n + 1) (5 + 4 n)
                                            /

                                                      2                    2
    (7 + 4 n) (9 + 4 n) (4 n + 11) (4 n + 1) (2 n + 3)  (2 n + 7) (2 n + 5)

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



                             and in Maple notation



9/32*(97984576*n^11+2020305808*n^10+18392244112*n^9+97512686328*n^8+
334201671840*n^7+776428755687*n^6+1245716742443*n^5+1377552520517*n^4+
1026405069389*n^3+489144217500*n^2+133561801800*n+15717240000)*(6*n+1)*(3*n+1)*
(3*n+2)*(6*n+5)/(2*n+1)/(5+4*n)/(7+4*n)/(9+4*n)/(4*n+11)/(4*n+1)/(2*n+3)^2/(2*n
+7)/(2*n+5)^2/(n+2)/(n+5)/(4*n+3)/(9+2*n)










Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n, 1 + n, n],  is



              10              9               8               7
54 (17632767 n   + 279805206 n  + 1880701804 n  + 7053920406 n

                    6                5                4                3
     + 16342034469 n  + 24398161596 n  + 23713610544 n  + 14769126456 n

                   2
     + 5620235552 n  + 1174420800 n + 101606400) (6 n + 1) (3 n + 1) (3 n + 2)

    (6 n + 5)/(5 (6 + 5 n) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 7) (5 n + 8)

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

    )



                             and in Maple notation



54/5*(17632767*n^10+279805206*n^9+1880701804*n^8+7053920406*n^7+16342034469*n^6
+24398161596*n^5+23713610544*n^4+14769126456*n^3+5620235552*n^2+1174420800*n+
101606400)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)/(6+5*n)/(5*n+2)/(5*n+3)/(5*n+4)/(5*n
+7)/(5*n+8)/(5*n+9)/(5*n+1)/(2*n+3)/(2*n+7)/(2*n+5)/(n+4)/(n+5)/(9+2*n)








Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, 1 + n, 1 + n, n],  is



9 (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5)

          4         3          2
    (416 n  + 4279 n  + 15256 n  + 21809 n + 10080)/(2 (2 n + 3) (2 n + 7)

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



                             and in Maple notation



9/2*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(416*n^4+4279*n^3+15256*n^2+21809*n+10080)/
(2*n+3)/(2*n+7)/(2*n+5)/(n+2)/(n+3)/(n+4)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, 1 + n, n + 2, n],  is



                                                             8             7
63 (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (1978261 n  + 32910881 n

                  6              5               4               3
     + 231421996 n  + 900604754 n  + 2126663725 n  + 3127144253 n

                   2
     + 2801568930 n  + 1400328000 n + 299376000)/(10 (5 n + 8) (5 n + 11)

    (6 + 5 n) (9 + 2 n) (2 n + 7) (2 n + 5) (2 n + 3) (5 n + 9) (5 n + 7)

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



                             and in Maple notation



63/10*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(1978261*n^8+32910881*n^7+231421996
*n^6+900604754*n^5+2126663725*n^4+3127144253*n^3+2801568930*n^2+1400328000*n+
299376000)/(5*n+8)/(5*n+11)/(6+5*n)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(5*n+9)/(5*
n+7)/(n+2)/(n+3)/(n+4)/(n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, 1 + n, n + 2, 1 + n],  is



                                                             6           5
324 (6 n + 7) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (8356 n  + 118133 n

               4            3            2
     + 669538 n  + 1950620 n  + 3087298 n  + 2521655 n + 831600) (3 n + 4)/(5

    (6 + 5 n) (5 n + 7) (5 n + 8) (5 n + 9) (5 n + 11) (2 n + 7) (2 n + 5)

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



                             and in Maple notation



324/5*(6*n+7)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(8356*n^6+118133*n^5+669538*n^4+
1950620*n^3+3087298*n^2+2521655*n+831600)*(3*n+4)/(6+5*n)/(5*n+7)/(5*n+8)/(5*n+
9)/(5*n+11)/(2*n+7)/(2*n+5)/(n+2)/(n+3)/(n+4)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 2, 1 + n, n],  is



              10             9              8               7               6
945 (1494600 n   + 34271328 n  + 347268136 n  + 2047736791 n  + 7780665935 n

                    5                4                3                2
     + 19900399897 n  + 34686230969 n  + 40665798544 n  + 30676189000 n

     + 13438173600 n + 2594592000) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5)

              /
    (1 + n)  /  (32 (5 + 4 n) (7 + 4 n) (9 + 4 n) (4 n + 11) (4 n + 13)
            /

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



                             and in Maple notation



945/32*(1494600*n^10+34271328*n^9+347268136*n^8+2047736791*n^7+7780665935*n^6+
19900399897*n^5+34686230969*n^4+40665798544*n^3+30676189000*n^2+13438173600*n+
2594592000)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(1+n)/(5+4*n)/(7+4*n)/(9+4*n)/(4*n+
11)/(4*n+13)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2)/(n+3)/(n+4)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 2, n + 2, n],  is



                     9              8               7               6
27 (1 + n) (8709982 n  + 181393805 n  + 1643662005 n  + 8506492761 n

                    5                4                3                2
     + 27711977817 n  + 58936300656 n  + 81834604364 n  + 71550439810 n

     + 35755030800 n + 7783776000) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1)

                /
    (6 n + 7)  /  (16 (5 + 4 n) (7 + 4 n) (9 + 4 n) (4 n + 11) (4 n + 13)
              /

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



                             and in Maple notation



27/16*(1+n)*(8709982*n^9+181393805*n^8+1643662005*n^7+8506492761*n^6+
27711977817*n^5+58936300656*n^4+81834604364*n^3+71550439810*n^2+35755030800*n+
7783776000)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(6*n+7)/(5+4*n)/(7+4*n)/(9+4*n)/(4*
n+11)/(4*n+13)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2)/(n+3)/(n+4)/(n+5)/(9+2*n)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 2, n + 2, 1 + n],  is



                                 9              8              7
27 (6 n + 7) (6 n + 1) (4909432 n  + 103914496 n  + 959300592 n

                   6                5                4                3
     + 5071684695 n  + 16929016428 n  + 37012575234 n  + 53024839808 n

                    2
     + 48020475815 n  + 24958633500 n + 5675670000) (3 n + 1) (3 n + 2)

                          /
    (6 n + 5) (3 n + 4)  /  (16 (5 + 4 n) (7 + 4 n) (9 + 4 n) (4 n + 11)
                        /

                        2                    2
    (4 n + 13) (2 n + 3)  (2 n + 7) (2 n + 5)  (n + 2) (n + 3) (n + 4) (n + 5)

    (9 + 2 n))



                             and in Maple notation



27/16*(6*n+7)*(6*n+1)*(4909432*n^9+103914496*n^8+959300592*n^7+5071684695*n^6+
16929016428*n^5+37012575234*n^4+53024839808*n^3+48020475815*n^2+24958633500*n+
5675670000)*(3*n+1)*(3*n+2)*(6*n+5)*(3*n+4)/(5+4*n)/(7+4*n)/(9+4*n)/(4*n+11)/(4
*n+13)/(2*n+3)^2/(2*n+7)/(2*n+5)^2/(n+2)/(n+3)/(n+4)/(n+5)/(9+2*n)










Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, 1 + n, 1 + n, n],  is



                                                            10              9
7 (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (9694527 n   + 248136778 n

                   8                7                6                 5
     + 2809456597 n  + 18510956258 n  + 78513461353 n  + 223723057382 n

                     4                 3                 2
     + 433142995243 n  + 561756853102 n  + 466273273560 n  + 223226323200 n

     + 46702656000)/(2 (3 n + 13) (10 + 3 n) (3 n + 7) (9 + 2 n) (2 n + 7)

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

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



                             and in Maple notation



7/2*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(9694527*n^10+248136778*n^9+
2809456597*n^8+18510956258*n^7+78513461353*n^6+223723057382*n^5+433142995243*n^
4+561756853102*n^3+466273273560*n^2+223226323200*n+46702656000)/(3*n+13)/(10+3*
n)/(3*n+7)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n+3)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+4)/(n+
2)/(n+3)/(n+4)/(n+5)








Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 2, 1 + n, n],  is



                                                                       9
10 (6 n + 7) (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (2339881 n

                 8              7               6               5
     + 52081873 n  + 505105840 n  + 2798997266 n  + 9757066957 n

                    4                3                2
     + 22164089777 n  + 32768999274 n  + 30365455852 n  + 15979428720 n

     + 3632428800)/((3 n + 13) (10 + 3 n) (3 n + 7) (9 + 2 n) (2 n + 7)

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

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



                             and in Maple notation



10*(6*n+7)*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(2339881*n^9+52081873*n^8+
505105840*n^7+2798997266*n^6+9757066957*n^5+22164089777*n^4+32768999274*n^3+
30365455852*n^2+15979428720*n+3632428800)/(3*n+13)/(10+3*n)/(3*n+7)/(9+2*n)/(2*
n+7)/(2*n+5)/(2*n+3)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+4)/(n+2)/(n+3)/(n+4)/(n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 2, n + 2, n],  is



                                                                      8
5 (6 n + 7) (1 + n) (6 n + 5) (3 n + 2) (3 n + 1) (6 n + 1) (2832264 n

                 7              6               5               4
     + 55841015 n  + 469841290 n  + 2201131934 n  + 6273526816 n

                    3                2
     + 11128414355 n  + 11987431246 n  + 7163103960 n + 1816214400)/((3 n + 13)

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

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



                             and in Maple notation



5*(6*n+7)*(1+n)*(6*n+5)*(3*n+2)*(3*n+1)*(6*n+1)*(2832264*n^8+55841015*n^7+
469841290*n^6+2201131934*n^5+6273526816*n^4+11128414355*n^3+11987431246*n^2+
7163103960*n+1816214400)/(3*n+13)/(10+3*n)/(3*n+7)/(9+2*n)/(2*n+7)/(2*n+5)/(2*n
+3)/(3*n+11)/(3*n+8)/(3*n+5)/(n+2)/(n+3)/(n+4)/(n+5)




Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 2, n + 2, 1 + n],  is



                                                             6          5
210 (6 n + 7) (6 n + 1) (3 n + 1) (3 n + 2) (6 n + 5) (3823 n  + 59909 n

               4            3            2
     + 377617 n  + 1220771 n  + 2126968 n  + 1887968 n + 665280) (5 n + 13)/(

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

    (n + 3) (n + 4) (n + 5) (3 n + 13) (9 + 2 n))



                             and in Maple notation



210*(6*n+7)*(6*n+1)*(3*n+1)*(3*n+2)*(6*n+5)*(3823*n^6+59909*n^5+377617*n^4+
1220771*n^3+2126968*n^2+1887968*n+665280)*(5*n+13)/(3*n+7)/(3*n+8)/(2*n+7)/(10+
3*n)/(2*n+5)/(3*n+11)/(n+2)/(n+3)/(n+4)/(n+5)/(3*n+13)/(9+2*n)




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



        This ends this article that took, 5949.761, seconds to produce.