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 By Shalosh B. Ekhad Let RF(a,n) be the raising factorial: a(a+1)...(a+n-1); Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, 1 + n], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n + 2], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n + 3], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n + 4], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n, n + 5], is n 256 RF(1/4, n) RF(1/2, n) RF(3/4, n) ------------------------------------- RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(1/4,n)*RF(1/2,n)*RF(3/4,n)/RF(4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, 1 + n], is n 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ----------------------------------------------------------- RF(5/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(5/3,n)/RF(7/3,n)/RF( 4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, n + 2], is n 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ----------------------------------------------------------- RF(5/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(5/3,n)/RF(7/3,n)/RF( 4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, n + 3], is n 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ----------------------------------------------------------- RF(5/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(5/3,n)/RF(7/3,n)/RF( 4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, n + 4], is n 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ----------------------------------------------------------- RF(5/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(5/3,n)/RF(7/3,n)/RF( 4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n, n + 5], is n 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ----------------------------------------------------------- RF(5/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(2, n) and in Maple notation 256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(5/3,n)/RF(7/3,n)/RF( 4,n)/RF(3,n)/RF(2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2, n + 2], is n 2 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ------------------------------------------------------------- RF(8/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(1, n) and in Maple notation 2*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(7/3,n)/ RF(4,n)/RF(3,n)/RF(1,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2, n + 3], is n 2 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ------------------------------------------------------------- RF(8/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(1, n) and in Maple notation 2*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(7/3,n)/ RF(4,n)/RF(3,n)/RF(1,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2, n + 4], is n 2 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ------------------------------------------------------------- RF(8/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(1, n) and in Maple notation 2*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(7/3,n)/ RF(4,n)/RF(3,n)/RF(1,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2, n + 5], is n 2 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) ------------------------------------------------------------- RF(8/3, n) RF(7/3, n) RF(4, n) RF(3, n) RF(1, n) and in Maple notation 2*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(7/3,n)/ RF(4,n)/RF(3,n)/RF(1,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 3, n + 3], is n 1/30 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 2 / 2 (103 n + 251 n + 150) / (RF(4, n) RF(7/3, n) RF(8/3, n) RF(3, n) ) / and in Maple notation 1/30*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(4,n)/RF(7/3,n)/ RF(8/3,n)/RF(3,n)^2*(103*n^2+251*n+150) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 3, n + 4], is n 1/30 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 2 / 2 (103 n + 251 n + 150) / (RF(4, n) RF(7/3, n) RF(8/3, n) RF(3, n) ) / and in Maple notation 1/30*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(4,n)/RF(7/3,n)/ RF(8/3,n)/RF(3,n)^2*(103*n^2+251*n+150) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 3, n + 5], is n 1/30 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 2 / 2 (103 n + 251 n + 150) / (RF(4, n) RF(7/3, n) RF(8/3, n) RF(3, n) ) / and in Maple notation 1/30*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(4,n)/RF(7/3,n)/ RF(8/3,n)/RF(3,n)^2*(103*n^2+251*n+150) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 4, n + 4], is n 1/105 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 3 2 / (541 n + 2316 n + 3233 n + 1470) / (RF(8/3, n) RF(10/3, n) RF(4, n) / 2 RF(3, n) ) and in Maple notation 1/105*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(10/3 ,n)/RF(4,n)/RF(3,n)^2*(541*n^3+2316*n^2+3233*n+1470) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 4, n + 5], is n 1/105 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 3 2 / (541 n + 2316 n + 3233 n + 1470) / (RF(8/3, n) RF(10/3, n) RF(4, n) / 2 RF(3, n) ) and in Maple notation 1/105*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(8/3,n)/RF(10/3 ,n)/RF(4,n)/RF(3,n)^2*(541*n^3+2316*n^2+3233*n+1470) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 5, n + 5], is n 1/1680 256 RF(5/4, n) RF(2/3, n) RF(3/2, n) RF(1/3, n) RF(7/4, n) 4 3 2 / (11531 n + 75122 n + 179449 n + 186058 n + 70560) / (RF(11/3, n) / 2 RF(10/3, n) RF(4, n) RF(3, n) ) and in Maple notation 1/1680*256^n*RF(5/4,n)*RF(2/3,n)*RF(3/2,n)*RF(1/3,n)*RF(7/4,n)/RF(11/3,n)/RF(10 /3,n)/RF(4,n)/RF(3,n)^2*(11531*n^4+75122*n^3+179449*n^2+186058*n+70560) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 1, n + 1], is n 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(5/4, n) ------------------------------------------------ RF(4, n) RF(3, n) RF(2, n) RF(7/2, n) and in Maple notation 256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(5/4,n)/RF(4,n)/RF(3,n)/RF(2,n)/RF(7/2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 1, n + 2], is n 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(5/4, n) ------------------------------------------------ RF(4, n) RF(3, n) RF(2, n) RF(7/2, n) and in Maple notation 256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(5/4,n)/RF(4,n)/RF(3,n)/RF(2,n)/RF(7/2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 1, n + 3], is n 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(5/4, n) ------------------------------------------------ RF(4, n) RF(3, n) RF(2, n) RF(7/2, n) and in Maple notation 256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(5/4,n)/RF(4,n)/RF(3,n)/RF(2,n)/RF(7/2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 1, n + 4], is n 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(5/4, n) ------------------------------------------------ RF(4, n) RF(3, n) RF(2, n) RF(7/2, n) and in Maple notation 256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(5/4,n)/RF(4,n)/RF(3,n)/RF(2,n)/RF(7/2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 1, n + 5], is n 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(5/4, n) ------------------------------------------------ RF(4, n) RF(3, n) RF(2, n) RF(7/2, n) and in Maple notation 256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(5/4,n)/RF(4,n)/RF(3,n)/RF(2,n)/RF(7/2,n) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 2, n + 2], is n 2 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(9/4, n) -------------------------------------------------- 2 RF(4, n) RF(7/2, n) RF(3, n) and in Maple notation 2*256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(9/4,n)/RF(4,n)/RF(7/2,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 2, n + 3], is n 2 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(9/4, n) -------------------------------------------------- 2 RF(4, n) RF(7/2, n) RF(3, n) and in Maple notation 2*256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(9/4,n)/RF(4,n)/RF(7/2,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 2, n + 4], is n 2 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(9/4, n) -------------------------------------------------- 2 RF(4, n) RF(7/2, n) RF(3, n) and in Maple notation 2*256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(9/4,n)/RF(4,n)/RF(7/2,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 2, n + 5], is n 2 256 RF(7/4, n) RF(3/2, n) RF(1/2, n) RF(9/4, n) -------------------------------------------------- 2 RF(4, n) RF(7/2, n) RF(3, n) and in Maple notation 2*256^n*RF(7/4,n)*RF(3/2,n)*RF(1/2,n)*RF(9/4,n)/RF(4,n)/RF(7/2,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 3, n + 3], is n 5 256 RF(11/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(9/4, n) 42 / / 25 RF(--, n) / |RF(8/3, n) RF(7/2, n) RF(10/3, n) RF(--, n) RF(4, n) 17 / \ 17 2\ RF(3, n) | / and in Maple notation 5*256^n*RF(11/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(9/4,n)*RF(42/17,n )/RF(8/3,n)/RF(7/2,n)/RF(10/3,n)/RF(25/17,n)/RF(4,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 3, n + 4], is n 5 256 RF(11/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(9/4, n) 42 / / 25 RF(--, n) / |RF(8/3, n) RF(7/2, n) RF(10/3, n) RF(--, n) RF(4, n) 17 / \ 17 2\ RF(3, n) | / and in Maple notation 5*256^n*RF(11/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(9/4,n)*RF(42/17,n )/RF(8/3,n)/RF(7/2,n)/RF(10/3,n)/RF(25/17,n)/RF(4,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 3, n + 5], is n 5 256 RF(11/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(9/4, n) 42 / / 25 RF(--, n) / |RF(8/3, n) RF(7/2, n) RF(10/3, n) RF(--, n) RF(4, n) 17 / \ 17 2\ RF(3, n) | / and in Maple notation 5*256^n*RF(11/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(9/4,n)*RF(42/17,n )/RF(8/3,n)/RF(7/2,n)/RF(10/3,n)/RF(25/17,n)/RF(4,n)/RF(3,n)^2 Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 4, n + 4], is n 1/10 256 RF(9/4, n) RF(4/3, n) RF(1/2, n) RF(5/2, n) RF(5/3, n) RF(11/4, n) 2 / (44 n + 159 n + 140) / (RF(11/3, n) RF(7/2, n) RF(10/3, n) RF(4, n) / 2 RF(3, n) ) and in Maple notation 1/10*256^n*RF(9/4,n)*RF(4/3,n)*RF(1/2,n)*RF(5/2,n)*RF(5/3,n)*RF(11/4,n)/RF(11/3 ,n)/RF(7/2,n)/RF(10/3,n)/RF(4,n)/RF(3,n)^2*(44*n^2+159*n+140) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 4, n + 5], is n 1/10 256 RF(9/4, n) RF(4/3, n) RF(1/2, n) RF(5/2, n) RF(5/3, n) RF(11/4, n) 2 / (44 n + 159 n + 140) / (RF(11/3, n) RF(7/2, n) RF(10/3, n) RF(4, n) / 2 RF(3, n) ) and in Maple notation 1/10*256^n*RF(9/4,n)*RF(4/3,n)*RF(1/2,n)*RF(5/2,n)*RF(5/3,n)*RF(11/4,n)/RF(11/3 ,n)/RF(7/2,n)/RF(10/3,n)/RF(4,n)/RF(3,n)^2*(44*n^2+159*n+140) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 1, n + 5, n + 5], is n 2/9 256 RF(11/4, n) RF(5/3, n) RF(1/2, n) RF(4/3, n) RF(9/4, n) 2 / 2 (58 n + 212 n + 189) / (RF(11/3, n) RF(10/3, n) RF(3, n) RF(4, n) ) / and in Maple notation 2/9*256^n*RF(11/4,n)*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)*RF(9/4,n)/RF(11/3,n)/RF(10/3 ,n)/RF(3,n)/RF(4,n)^2*(58*n^2+212*n+189) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2, n + 2], is n 1/210 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(7/4, n) 3 2 / (269 n + 1314 n + 2071 n + 1050) / (RF(8/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(3, n) RF(4, n) ) and in Maple notation 1/210*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/4,n)/RF(8/3, n)/RF(7/2,n)/RF(10/3,n)/RF(3,n)/RF(4,n)^2*(269*n^3+1314*n^2+2071*n+1050) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2, n + 3], is n 1/210 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(7/4, n) 3 2 / (269 n + 1314 n + 2071 n + 1050) / (RF(8/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(3, n) RF(4, n) ) and in Maple notation 1/210*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/4,n)/RF(8/3, n)/RF(7/2,n)/RF(10/3,n)/RF(3,n)/RF(4,n)^2*(269*n^3+1314*n^2+2071*n+1050) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2, n + 4], is n 1/210 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(7/4, n) 3 2 / (269 n + 1314 n + 2071 n + 1050) / (RF(8/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(3, n) RF(4, n) ) and in Maple notation 1/210*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/4,n)/RF(8/3, n)/RF(7/2,n)/RF(10/3,n)/RF(3,n)/RF(4,n)^2*(269*n^3+1314*n^2+2071*n+1050) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2, n + 5], is n 1/210 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(7/4, n) 3 2 / (269 n + 1314 n + 2071 n + 1050) / (RF(8/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(3, n) RF(4, n) ) and in Maple notation 1/210*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(7/4,n)/RF(8/3, n)/RF(7/2,n)/RF(10/3,n)/RF(3,n)/RF(4,n)^2*(269*n^3+1314*n^2+2071*n+1050) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 3, n + 3], is n 3/8 256 RF(14/5, n) RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) 2 / RF(11/4, n) (13 n + 43 n + 32) / (RF(11/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(9/5, n) RF(3, n) RF(4, n) ) and in Maple notation 3/8*256^n*RF(14/5,n)*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4, n)/RF(11/3,n)/RF(7/2,n)/RF(10/3,n)/RF(9/5,n)/RF(3,n)/RF(4,n)^2*(13*n^2+43*n+32) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 3, n + 4], is n 3/8 256 RF(14/5, n) RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) 2 / RF(11/4, n) (13 n + 43 n + 32) / (RF(11/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(9/5, n) RF(3, n) RF(4, n) ) and in Maple notation 3/8*256^n*RF(14/5,n)*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4, n)/RF(11/3,n)/RF(7/2,n)/RF(10/3,n)/RF(9/5,n)/RF(3,n)/RF(4,n)^2*(13*n^2+43*n+32) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 3, n + 5], is n 3/8 256 RF(14/5, n) RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) 2 / RF(11/4, n) (13 n + 43 n + 32) / (RF(11/3, n) RF(7/2, n) RF(10/3, n) / 2 RF(9/5, n) RF(3, n) RF(4, n) ) and in Maple notation 3/8*256^n*RF(14/5,n)*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4, n)/RF(11/3,n)/RF(7/2,n)/RF(10/3,n)/RF(9/5,n)/RF(3,n)/RF(4,n)^2*(13*n^2+43*n+32) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 4, n + 4], is n 1/1080 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 4 3 2 / (3289 n + 24783 n + 68555 n + 82101 n + 35640) / (RF(11/3, n) / 3 RF(7/2, n) RF(10/3, n) RF(4, n) ) and in Maple notation 1/1080*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(7/2,n)/RF(10/3,n)/RF(4,n)^3*(3289*n^4+24783*n^3+68555*n^2+82101*n+ 35640) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 4, n + 5], is n 1/1080 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 4 3 2 / (3289 n + 24783 n + 68555 n + 82101 n + 35640) / (RF(11/3, n) / 3 RF(7/2, n) RF(10/3, n) RF(4, n) ) and in Maple notation 1/1080*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(7/2,n)/RF(10/3,n)/RF(4,n)^3*(3289*n^4+24783*n^3+68555*n^2+82101*n+ 35640) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 5, n + 5], is n 1/5400 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 5 4 3 2 / (17083 n + 177251 n + 722315 n + 1441177 n + 1402350 n + 529200) / ( / 3 RF(11/3, n) RF(7/2, n) RF(13/3, n) RF(4, n) ) and in Maple notation 1/5400*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(7/2,n)/RF(13/3,n)/RF(4,n)^3*(17083*n^5+177251*n^4+722315*n^3+1441177*n ^2+1402350*n+529200) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 3, n + 3], is n 1/3780 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 5 4 3 2 / (6652 n + 65231 n + 251414 n + 475303 n + 439728 n + 158760) / ( / 3 RF(11/3, n) RF(9/2, n) RF(10/3, n) RF(4, n) ) and in Maple notation 1/3780*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(9/2,n)/RF(10/3,n)/RF(4,n)^3*(6652*n^5+65231*n^4+251414*n^3+475303*n^2+ 439728*n+158760) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 3, n + 4], is n 1/3780 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 5 4 3 2 / (6652 n + 65231 n + 251414 n + 475303 n + 439728 n + 158760) / ( / 3 RF(11/3, n) RF(9/2, n) RF(10/3, n) RF(4, n) ) and in Maple notation 1/3780*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(9/2,n)/RF(10/3,n)/RF(4,n)^3*(6652*n^5+65231*n^4+251414*n^3+475303*n^2+ 439728*n+158760) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 3, n + 5], is n 1/3780 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 5 4 3 2 / (6652 n + 65231 n + 251414 n + 475303 n + 439728 n + 158760) / ( / 3 RF(11/3, n) RF(9/2, n) RF(10/3, n) RF(4, n) ) and in Maple notation 1/3780*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF(11 /3,n)/RF(9/2,n)/RF(10/3,n)/RF(4,n)^3*(6652*n^5+65231*n^4+251414*n^3+475303*n^2+ 439728*n+158760) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 4, n + 4], is n 1/18900 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 6 5 4 3 2 (32296 n + 403358 n + 2071127 n + 5588824 n + 8344749 n + 6522390 n / 3 + 2079000) / (RF(11/3, n) RF(9/2, n) RF(13/3, n) RF(4, n) ) / and in Maple notation 1/18900*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 11/3,n)/RF(9/2,n)/RF(13/3,n)/RF(4,n)^3*(32296*n^6+403358*n^5+2071127*n^4+ 5588824*n^3+8344749*n^2+6522390*n+2079000) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 4, n + 5], is n 1/18900 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 6 5 4 3 2 (32296 n + 403358 n + 2071127 n + 5588824 n + 8344749 n + 6522390 n / 3 + 2079000) / (RF(11/3, n) RF(9/2, n) RF(13/3, n) RF(4, n) ) / and in Maple notation 1/18900*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 11/3,n)/RF(9/2,n)/RF(13/3,n)/RF(4,n)^3*(32296*n^6+403358*n^5+2071127*n^4+ 5588824*n^3+8344749*n^2+6522390*n+2079000) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 5, n + 5], is n 1/207900 256 RF(9/4, n) RF(5/3, n) RF(5/2, n) RF(1/2, n) RF(4/3, n) 7 6 5 4 RF(11/4, n) (327904 n + 5120560 n + 33842742 n + 122594717 n 3 2 / + 262565798 n + 331979847 n + 229001580 n + 66320100) / (RF(14/3, n) / 3 RF(9/2, n) RF(13/3, n) RF(4, n) ) and in Maple notation 1/207900*256^n*RF(9/4,n)*RF(5/3,n)*RF(5/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 14/3,n)/RF(9/2,n)/RF(13/3,n)/RF(4,n)^3*(327904*n^7+5120560*n^6+33842742*n^5+ 122594717*n^4+262565798*n^3+331979847*n^2+229001580*n+66320100) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 4, n + 4, n + 4], is n 1/83160 256 RF(9/4, n) RF(5/3, n) RF(7/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 7 6 5 4 3 (168248 n + 2686492 n + 18123759 n + 66901710 n + 145772677 n 2 / + 187214918 n + 130981956 n + 38419920) / (RF(14/3, n) RF(9/2, n) / 2 RF(13/3, n) RF(5, n) RF(4, n) ) and in Maple notation 1/83160*256^n*RF(9/4,n)*RF(5/3,n)*RF(7/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 14/3,n)/RF(9/2,n)/RF(13/3,n)/RF(5,n)/RF(4,n)^2*(168248*n^7+2686492*n^6+18123759 *n^5+66901710*n^4+145772677*n^3+187214918*n^2+130981956*n+38419920) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 4, n + 4, n + 5], is n 1/83160 256 RF(9/4, n) RF(5/3, n) RF(7/2, n) RF(1/2, n) RF(4/3, n) RF(11/4, n) 7 6 5 4 3 (168248 n + 2686492 n + 18123759 n + 66901710 n + 145772677 n 2 / + 187214918 n + 130981956 n + 38419920) / (RF(14/3, n) RF(9/2, n) / 2 RF(13/3, n) RF(5, n) RF(4, n) ) and in Maple notation 1/83160*256^n*RF(9/4,n)*RF(5/3,n)*RF(7/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 14/3,n)/RF(9/2,n)/RF(13/3,n)/RF(5,n)/RF(4,n)^2*(168248*n^7+2686492*n^6+18123759 *n^5+66901710*n^4+145772677*n^3+187214918*n^2+130981956*n+38419920) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 4, n + 5, n + 5], is n 1/997920 256 RF(9/4, n) RF(5/3, n) RF(7/2, n) RF(1/2, n) RF(4/3, n) 8 7 6 5 RF(11/4, n) (1639376 n + 31727128 n + 265249634 n + 1250187037 n 4 3 2 + 3629870384 n + 6640602667 n + 7464915126 n + 4706349768 n / 2 + 1271350080) / (RF(14/3, n) RF(9/2, n) RF(13/3, n) RF(4, n) RF(5, n) ) / and in Maple notation 1/997920*256^n*RF(9/4,n)*RF(5/3,n)*RF(7/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF( 14/3,n)/RF(9/2,n)/RF(13/3,n)/RF(4,n)/RF(5,n)^2*(1639376*n^8+31727128*n^7+ 265249634*n^6+1250187037*n^5+3629870384*n^4+6640602667*n^3+7464915126*n^2+ 4706349768*n+1271350080) Theorem: The number of 4-rowed Truncated Standard Young Tableaux of shape, [n, n + 5, n + 5, n + 5], is n 1/5559840 256 RF(9/4, n) RF(5/3, n) RF(9/2, n) RF(1/2, n) RF(4/3, n) 9 8 7 6 RF(11/4, n) (11538424 n + 266422180 n + 2700801350 n + 15767135395 n 5 4 3 2 + 58376817827 n + 142031047855 n + 226851642775 n + 229080620730 n / + 132515328024 n + 33392399040) / (RF(14/3, n) RF(11/2, n) RF(16/3, n) / 2 RF(4, n) RF(5, n) ) and in Maple notation 1/5559840*256^n*RF(9/4,n)*RF(5/3,n)*RF(9/2,n)*RF(1/2,n)*RF(4/3,n)*RF(11/4,n)/RF (14/3,n)/RF(11/2,n)/RF(16/3,n)/RF(4,n)/RF(5,n)^2*(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) ------------------------- This ends this article that took, 17.249, seconds to produce.