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 By Shalosh B. Ekhad Let RF(a,n) be the raising factorial: a(a+1)...(a+n-1); Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, n, n], is n 46656 RF(5/6, n) RF(2/3, n) RF(1/2, n) RF(1/3, n) RF(1/6, n) ------------------------------------------------------------- RF(5, n) RF(4, n) RF(3, n) RF(2, n) RF(6, n) and in Maple notation 46656^n*RF(5/6,n)*RF(2/3,n)*RF(1/2,n)*RF(1/3,n)*RF(1/6,n)/RF(5,n)/RF(4,n)/RF(3, n)/RF(2,n)/RF(6,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, n, 1 + n], is n 46656 RF(5/6, n) RF(2/3, n) RF(1/2, n) RF(1/3, n) RF(1/6, n) ------------------------------------------------------------- RF(5, n) RF(4, n) RF(3, n) RF(2, n) RF(6, n) and in Maple notation 46656^n*RF(5/6,n)*RF(2/3,n)*RF(1/2,n)*RF(1/3,n)*RF(1/6,n)/RF(5,n)/RF(4,n)/RF(3, n)/RF(2,n)/RF(6,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, n, n + 2], is n 46656 RF(5/6, n) RF(2/3, n) RF(1/2, n) RF(1/3, n) RF(1/6, n) ------------------------------------------------------------- RF(5, n) RF(4, n) RF(3, n) RF(2, n) RF(6, n) and in Maple notation 46656^n*RF(5/6,n)*RF(2/3,n)*RF(1/2,n)*RF(1/3,n)*RF(1/6,n)/RF(5,n)/RF(4,n)/RF(3, n)/RF(2,n)/RF(6,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, 1 + n, 1 + n], is n 46656 RF(11/6, n) RF(2/5, n) RF(5/3, n) RF(4/5, n) RF(3/2, n) RF(1/5, n) RF(4/3, n) RF(3/5, n) RF(7/6, n)/(RF(9/5, n) RF(8/5, n) RF(7/5, n) RF(11/5, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 46656^n*RF(11/6,n)*RF(2/5,n)*RF(5/3,n)*RF(4/5,n)*RF(3/2,n)*RF(1/5,n)*RF(4/3,n)* RF(3/5,n)*RF(7/6,n)/RF(9/5,n)/RF(8/5,n)/RF(7/5,n)/RF(11/5,n)/RF(6,n)/RF(5,n)/RF (4,n)/RF(3,n)/RF(2,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, 1 + n, n + 2], is n 46656 RF(11/6, n) RF(2/5, n) RF(5/3, n) RF(4/5, n) RF(3/2, n) RF(1/5, n) RF(4/3, n) RF(3/5, n) RF(7/6, n)/(RF(9/5, n) RF(8/5, n) RF(7/5, n) RF(11/5, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 46656^n*RF(11/6,n)*RF(2/5,n)*RF(5/3,n)*RF(4/5,n)*RF(3/2,n)*RF(1/5,n)*RF(4/3,n)* RF(3/5,n)*RF(7/6,n)/RF(9/5,n)/RF(8/5,n)/RF(7/5,n)/RF(11/5,n)/RF(6,n)/RF(5,n)/RF (4,n)/RF(3,n)/RF(2,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n, n + 2, n + 2], is n 2 46656 RF(15/8, n) RF(11/6, n) RF(2/5, n) RF(5/3, n) RF(4/5, n) RF(3/2, n) RF(1/5, n) RF(4/3, n) RF(3/5, n) RF(7/6, n)/(RF(7/8, n) RF(12/5, n) RF(9/5, n) RF(11/5, n) RF(8/5, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 2*46656^n*RF(15/8,n)*RF(11/6,n)*RF(2/5,n)*RF(5/3,n)*RF(4/5,n)*RF(3/2,n)*RF(1/5, n)*RF(4/3,n)*RF(3/5,n)*RF(7/6,n)/RF(7/8,n)/RF(12/5,n)/RF(9/5,n)/RF(11/5,n)/RF(8 /5,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, 1 + n, 1 + n, 1 + n], is n 1/15120 46656 RF(5/6, n) RF(1/4, n) RF(2/3, n) RF(3/2, n) RF(1/2, n) RF(4/3, n) RF(3/4, n) RF(7/6, n) 4 3 2 (9763 n + 47673 n + 82520 n + 59424 n + 15120)/(RF(11/4, n) RF(5/2, n) RF(13/4, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/15120*46656^n*RF(5/6,n)*RF(1/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/2,n)*RF(4/3,n)*RF( 3/4,n)*RF(7/6,n)/RF(11/4,n)/RF(5/2,n)/RF(13/4,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n )/RF(2,n)*(9763*n^4+47673*n^3+82520*n^2+59424*n+15120) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, 1 + n, 1 + n, n + 2], is n 1/15120 46656 RF(5/6, n) RF(1/4, n) RF(2/3, n) RF(3/2, n) RF(1/2, n) RF(4/3, n) RF(3/4, n) RF(7/6, n) 4 3 2 (9763 n + 47673 n + 82520 n + 59424 n + 15120)/(RF(11/4, n) RF(5/2, n) RF(13/4, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/15120*46656^n*RF(5/6,n)*RF(1/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/2,n)*RF(4/3,n)*RF( 3/4,n)*RF(7/6,n)/RF(11/4,n)/RF(5/2,n)/RF(13/4,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n )/RF(2,n)*(9763*n^4+47673*n^3+82520*n^2+59424*n+15120) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, 1 + n, n + 2, n + 2], is n 1/252 46656 RF(11/6, n) RF(1/4, n) RF(5/3, n) RF(3/2, n) RF(1/2, n) RF(4/3, n) 2 RF(3/4, n) RF(7/6, n) (277 n + 779 n + 504)/(RF(11/4, n) RF(5/2, n) RF(13/4, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/252*46656^n*RF(11/6,n)*RF(1/4,n)*RF(5/3,n)*RF(3/2,n)*RF(1/2,n)*RF(4/3,n)*RF(3 /4,n)*RF(7/6,n)/RF(11/4,n)/RF(5/2,n)/RF(13/4,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n) /RF(2,n)*(277*n^2+779*n+504) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n + 2, n + 2, n + 2], is n 91 5 46656 RF(11/6, n) RF(3/4, n) RF(5/3, n) RF(3/2, n) RF(1/2, n) RF(--, n) 31 / / 60 RF(4/3, n) RF(1/4, n) RF(7/6, n) RF(2, n) / |RF(13/4, n) RF(--, n) / \ 31 2\ RF(7/2, n) RF(11/4, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(1, n) | / and in Maple notation 5*46656^n*RF(11/6,n)*RF(3/4,n)*RF(5/3,n)*RF(3/2,n)*RF(1/2,n)*RF(91/31,n)*RF(4/3 ,n)*RF(1/4,n)*RF(7/6,n)*RF(2,n)/RF(13/4,n)/RF(60/31,n)/RF(7/2,n)/RF(11/4,n)/RF( 6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(1,n)^2 Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, 1 + n, 1 + n, 1 + n], is n 1/600 46656 RF(5/6, n) RF(5/3, n) RF(2/3, n) RF(3/2, n) RF(4/3, n) RF(1/3, n) 3 2 RF(7/6, n) (151 n + 809 n + 1300 n + 600)/(RF(11/3, n) RF(13/3, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/600*46656^n*RF(5/6,n)*RF(5/3,n)*RF(2/3,n)*RF(3/2,n)*RF(4/3,n)*RF(1/3,n)*RF(7/ 6,n)/RF(11/3,n)/RF(13/3,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n)*(151*n^3+809 *n^2+1300*n+600) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, 1 + n, 1 + n, n + 2], is n 1/600 46656 RF(5/6, n) RF(5/3, n) RF(2/3, n) RF(3/2, n) RF(4/3, n) RF(1/3, n) 3 2 RF(7/6, n) (151 n + 809 n + 1300 n + 600)/(RF(11/3, n) RF(13/3, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/600*46656^n*RF(5/6,n)*RF(5/3,n)*RF(2/3,n)*RF(3/2,n)*RF(4/3,n)*RF(1/3,n)*RF(7/ 6,n)/RF(11/3,n)/RF(13/3,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n)*(151*n^3+809 *n^2+1300*n+600) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, 1 + n, n + 2, n + 2], is n 1/21600 46656 RF(11/6, n) RF(7/5, n) RF(5/3, n) RF(2/3, n) RF(4/5, n) RF(3/2, n) RF(6/5, n) RF(4/3, n) RF(1/3, n) RF(13/6, n) 5 4 3 2 (6589 n + 52314 n + 158795 n + 231102 n + 161400 n + 43200)/( RF(12/5, n) RF(11/3, n) RF(14/5, n) RF(11/5, n) RF(13/3, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n)) and in Maple notation 1/21600*46656^n*RF(11/6,n)*RF(7/5,n)*RF(5/3,n)*RF(2/3,n)*RF(4/5,n)*RF(3/2,n)*RF (6/5,n)*RF(4/3,n)*RF(1/3,n)*RF(13/6,n)/RF(12/5,n)/RF(11/3,n)/RF(14/5,n)/RF(11/5 ,n)/RF(13/3,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n)*(6589*n^5+52314*n^4+ 158795*n^3+231102*n^2+161400*n+43200) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, n + 2, n + 2, n + 2], is n 1/20 46656 RF(11/6, n) RF(5/3, n) RF(2/3, n) RF(3/2, n) RF(4/3, n) RF(1/3, n) 2 / RF(13/6, n) (41 n + 134 n + 100) / (RF(4, n) RF(5, n) RF(6, n) / 2 RF(13/3, n) RF(11/3, n) RF(3, n) ) and in Maple notation 1/20*46656^n*RF(11/6,n)*RF(5/3,n)*RF(2/3,n)*RF(3/2,n)*RF(4/3,n)*RF(1/3,n)*RF(13 /6,n)/RF(4,n)/RF(5,n)/RF(6,n)/RF(13/3,n)/RF(11/3,n)/RF(3,n)^2*(41*n^2+134*n+100 ) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2, n + 2, n + 2, n + 2], is n 14 46656 RF(11/6, n) RF(5/3, n) RF(2/3, n) RF(3/2, n) RF(7/3, n) RF(1/3, n) RF(13/6, n)/(RF(14/3, n) RF(13/3, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(1, n)) and in Maple notation 14*46656^n*RF(11/6,n)*RF(5/3,n)*RF(2/3,n)*RF(3/2,n)*RF(7/3,n)*RF(1/3,n)*RF(13/6 ,n)/RF(14/3,n)/RF(13/3,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(1,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, 1 + n, 1 + n, 1 + n, 1 + n], is n 46656 RF(11/6, n) RF(5/3, n) RF(3/2, n) RF(1/2, n) RF(4/3, n) RF(7/6, n) ------------------------------------------------------------------------- RF(11/2, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 46656^n*RF(11/6,n)*RF(5/3,n)*RF(3/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/6,n)/RF(11/2,n) /RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, 1 + n, 1 + n, 1 + n, n + 2], is n 46656 RF(11/6, n) RF(5/3, n) RF(3/2, n) RF(1/2, n) RF(4/3, n) RF(7/6, n) ------------------------------------------------------------------------- RF(11/2, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 46656^n*RF(11/6,n)*RF(5/3,n)*RF(3/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/6,n)/RF(11/2,n) /RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, 1 + n, 1 + n, n + 2, n + 2], is n 2 46656 RF(11/6, n) RF(7/5, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(6/5, n) 33 / / RF(7/3, n) RF(--, n) RF(8/5, n) RF(13/6, n) / |RF(12/5, n) RF(11/2, n) 13 / \ 20 2\ RF(11/5, n) RF(--, n) RF(13/5, n) RF(6, n) RF(5, n) RF(4, n) RF(3, n) | 13 / and in Maple notation 2*46656^n*RF(11/6,n)*RF(7/5,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(6/5,n)*RF(7/3,n )*RF(33/13,n)*RF(8/5,n)*RF(13/6,n)/RF(12/5,n)/RF(11/2,n)/RF(11/5,n)/RF(20/13,n) /RF(13/5,n)/RF(6,n)/RF(5,n)/RF(4,n)/RF(3,n)^2 Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, 1 + n, n + 2, n + 2, n + 2], is n 1/11340 46656 RF(11/6, n) RF(5/4, n) RF(5/3, n) RF(1/2, n) RF(7/3, n) 2 RF(7/4, n) RF(13/6, n) RF(3/2, n) 5 4 3 2 / (2594 n + 24685 n + 92692 n + 171683 n + 156942 n + 56700) / ( / RF(13/4, n) RF(11/2, n) RF(7/2, n) RF(11/4, n) RF(6, n) RF(5, n) RF(3, n) 2 RF(4, n) ) and in Maple notation 1/11340*46656^n*RF(11/6,n)*RF(5/4,n)*RF(5/3,n)*RF(1/2,n)*RF(7/3,n)*RF(7/4,n)*RF (13/6,n)*RF(3/2,n)^2/RF(13/4,n)/RF(11/2,n)/RF(7/2,n)/RF(11/4,n)/RF(6,n)/RF(5,n) /RF(3,n)/RF(4,n)^2*(2594*n^5+24685*n^4+92692*n^3+171683*n^2+156942*n+56700) Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, n + 2, n + 2, n + 2, n + 2], is n 14 46656 RF(11/6, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(7/3, n) RF(13/6, n) / 2 / (RF(11/2, n) RF(6, n) RF(4, n) RF(3, n) RF(5, n) ) / and in Maple notation 14*46656^n*RF(11/6,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(7/3,n)*RF(13/6,n)/RF(11/ 2,n)/RF(6,n)/RF(4,n)/RF(3,n)/RF(5,n)^2 Theorem: The number of 6-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2, n + 2, n + 2, n + 2], is n 1/15840 46656 RF(11/6, n) RF(8/3, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) 6 5 RF(7/3, n) RF(4/3, n) RF(18/5, n) RF(13/6, n) (3823 n + 59909 n 4 3 2 / + 377617 n + 1220771 n + 2126968 n + 1887968 n + 665280) / ( / RF(14/3, n) RF(11/2, n) RF(16/3, n) RF(13/5, n) RF(5, n) RF(4, n) RF(3, n) 2 RF(6, n) ) and in Maple notation 1/15840*46656^n*RF(11/6,n)*RF(8/3,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(7/3,n)*RF (4/3,n)*RF(18/5,n)*RF(13/6,n)/RF(14/3,n)/RF(11/2,n)/RF(16/3,n)/RF(13/5,n)/RF(5, n)/RF(4,n)/RF(3,n)/RF(6,n)^2*(3823*n^6+59909*n^5+377617*n^4+1220771*n^3+2126968 *n^2+1887968*n+665280) ------------------------- This ends this article that took, 5551.327, seconds to produce.