Conjectured Explicit Formulas Enumerating 3-rowed Truncated Standard Young \

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



                              By Shalosh B. Ekhad





Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2],  is  the number of SYT of shape, [n, n, n], times



                                              2
                   3 (3 n + 2) (3 n + 1) (25 n  + 72 n + 50)
                   -----------------------------------------
                    2 (n + 2) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



3/2*(3*n+2)*(3*n+1)*(25*n^2+72*n+50)/(n+2)/(2*n+5)/(2*n+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n],  is  the number of SYT of shape, [n, n, n], times



                                                           2
                   45 (3 n + 4) (3 n + 2) (3 n + 1) (1 + n)
                -----------------------------------------------
                4 (n + 3) (n + 2) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



45/4*(3*n+4)*(3*n+2)*(3*n+1)*(1+n)^2/(n+3)/(n+2)/(2*n+5)/(2*n+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n],  is  the number of SYT of shape, [n, n, n], times



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



                             and in Maple notation



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




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2],  is  the number of SYT of shape, [n, n, n], times



                                              2
                  3 (3 n + 2) (3 n + 1) (131 n  + 383 n + 270)
                  --------------------------------------------
                    4 (n + 3) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



3/4*(3*n+2)*(3*n+1)*(131*n^2+383*n+270)/(n+3)/(2*n+5)/(2*n+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3],  is  the number of SYT of shape, [n, n, n], times



                                         3        2
             3 (3 n + 2) (3 n + 1) (181 n  + 939 n  + 1568 n + 840)
             ------------------------------------------------------
                4 (n + 3) (n + 2) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



3/4*(3*n+2)*(3*n+1)*(181*n^3+939*n^2+1568*n+840)/(n+3)/(n+2)/(2*n+5)/(2*n+1)/(3
+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n],  is  the number of SYT of shape, [n, n, n], times



         9 (3 n + 5) (3 n + 4) (1 + n) (3 n + 2) (3 n + 1) (13 n + 14)
         -------------------------------------------------------------
           8 (n + 3) (n + 2) (2 n + 7) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



9/8*(3*n+5)*(3*n+4)*(1+n)*(3*n+2)*(3*n+1)*(13*n+14)/(n+3)/(n+2)/(2*n+7)/(2*n+5)
/(2*n+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n],  is  the number of SYT of shape, [n, n, n], times



              45 (3 n + 5) (3 n + 4) (3 n + 2) (3 n + 1) (5 n + 7)
              ----------------------------------------------------
               8 (n + 3) (2 n + 7) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



45/8*(3*n+5)*(3*n+4)*(3*n+2)*(3*n+1)*(5*n+7)/(n+3)/(2*n+7)/(2*n+5)/(2*n+1)/(3+2
*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2],  is  the number of SYT of shape, [n, n, n], times



                                        3         2
           3 (3 n + 2) (3 n + 1) (1199 n  + 6336 n  + 10777 n + 5880)
           ----------------------------------------------------------
               8 (n + 3) (2 n + 7) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



3/8*(3*n+2)*(3*n+1)*(1199*n^3+6336*n^2+10777*n+5880)/(n+3)/(2*n+7)/(2*n+5)/(2*n
+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3],  is  the number of SYT of shape, [n, n, n], times



                                  4         3          2
      9 (3 n + 2) (3 n + 1) (641 n  + 5008 n  + 14289 n  + 17582 n + 7840)
      --------------------------------------------------------------------
           8 (n + 3) (n + 2) (2 n + 7) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



9/8*(3*n+2)*(3*n+1)*(641*n^4+5008*n^3+14289*n^2+17582*n+7840)/(n+3)/(n+2)/(2*n+
7)/(2*n+5)/(2*n+1)/(3+2*n)




Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4],  is  the number of SYT of shape, [n, n, n], times



                                 4          3          2
    3 (3 n + 2) (3 n + 1) (2647 n  + 21314 n  + 62285 n  + 78058 n + 35280)
    -----------------------------------------------------------------------
           8 (n + 3) (n + 2) (2 n + 7) (2 n + 5) (2 n + 1) (3 + 2 n)



                             and in Maple notation



3/8*(3*n+2)*(3*n+1)*(2647*n^4+21314*n^3+62285*n^2+78058*n+35280)/(n+3)/(n+2)/(2
*n+7)/(2*n+5)/(2*n+1)/(3+2*n)




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



         This ends this article that took, 3.551, seconds to produce.