Conjectured Explicit Formulas Enumerating 3-rowed Truncated Standard Young \ Tableau of shape [n,n+a1,n+a2] for a from 0<=a1<=a2<=, 5 By Shalosh B. Ekhad Let RF(a,n) be the raising factorial: a(a+1)...(a+n-1); Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, n], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, 1 + n], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 2], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 3], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 4], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n, n + 5], is n 27 RF(2/3, n) RF(1/3, n) ------------------------- RF(3, n) RF(2, n) and in Maple notation 27^n*RF(2/3,n)*RF(1/3,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, 1 + n], is n 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) ------------------------------------ RF(5/2, n) RF(3, n) RF(2, n) and in Maple notation 27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, n + 2], is n 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) ------------------------------------ RF(5/2, n) RF(3, n) RF(2, n) and in Maple notation 27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, n + 3], is n 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) ------------------------------------ RF(5/2, n) RF(3, n) RF(2, n) and in Maple notation 27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, n + 4], is n 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) ------------------------------------ RF(5/2, n) RF(3, n) RF(2, n) and in Maple notation 27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, 1 + n, n + 5], is n 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) ------------------------------------ RF(5/2, n) RF(3, n) RF(2, n) and in Maple notation 27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(3,n)/RF(2,n) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 2], is n 2 27 RF(15/7, n) RF(5/3, n) RF(1/2, n) RF(4/3, n) -------------------------------------------------- 2 RF(5/2, n) RF(8/7, n) RF(3, n) and in Maple notation 2*27^n*RF(15/7,n)*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(8/7,n)/RF(3,n)^2 Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 3], is n 2 27 RF(15/7, n) RF(5/3, n) RF(1/2, n) RF(4/3, n) -------------------------------------------------- 2 RF(5/2, n) RF(8/7, n) RF(3, n) and in Maple notation 2*27^n*RF(15/7,n)*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(8/7,n)/RF(3,n)^2 Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 4], is n 2 27 RF(15/7, n) RF(5/3, n) RF(1/2, n) RF(4/3, n) -------------------------------------------------- 2 RF(5/2, n) RF(8/7, n) RF(3, n) and in Maple notation 2*27^n*RF(15/7,n)*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(8/7,n)/RF(3,n)^2 Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 2, n + 5], is n 2 27 RF(15/7, n) RF(5/3, n) RF(1/2, n) RF(4/3, n) -------------------------------------------------- 2 RF(5/2, n) RF(8/7, n) RF(3, n) and in Maple notation 2*27^n*RF(15/7,n)*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(5/2,n)/RF(8/7,n)/RF(3,n)^2 Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 3], is n 2 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) (25 n + 72 n + 50) 1/10 -------------------------------------------------------- 2 RF(7/2, n) RF(3, n) and in Maple notation 1/10*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(7/2,n)/RF(3,n)^2*(25*n^2+72*n+50) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 4], is n 2 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) (25 n + 72 n + 50) 1/10 -------------------------------------------------------- 2 RF(7/2, n) RF(3, n) and in Maple notation 1/10*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(7/2,n)/RF(3,n)^2*(25*n^2+72*n+50) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 3, n + 5], is n 2 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) (25 n + 72 n + 50) 1/10 -------------------------------------------------------- 2 RF(7/2, n) RF(3, n) and in Maple notation 1/10*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(7/2,n)/RF(3,n)^2*(25*n^2+72*n+50) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 4, n + 4], is n 3 2 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) (181 n + 939 n + 1568 n + 840) 1/60 --------------------------------------------------------------------- RF(7/2, n) RF(4, n) RF(3, n) and in Maple notation 1/60*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(7/2,n)/RF(4,n)/RF(3,n)*(181*n^3+939* n^2+1568*n+840) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 4, n + 5], is n 3 2 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) (181 n + 939 n + 1568 n + 840) 1/60 --------------------------------------------------------------------- RF(7/2, n) RF(4, n) RF(3, n) and in Maple notation 1/60*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(7/2,n)/RF(4,n)/RF(3,n)*(181*n^3+939* n^2+1568*n+840) Theorem: The number of 3-rowed Truncated Standard Young Tableaux of shape, [n, n + 5, n + 5], is n 1/840 27 RF(5/3, n) RF(1/2, n) RF(4/3, n) 4 3 2 (2647 n + 21314 n + 62285 n + 78058 n + 35280)/(RF(9/2, n) RF(4, n) RF(3, n)) and in Maple notation 1/840*27^n*RF(5/3,n)*RF(1/2,n)*RF(4/3,n)/RF(9/2,n)/RF(4,n)/RF(3,n)*(2647*n^4+ 21314*n^3+62285*n^2+78058*n+35280) ------------------------- This ends this article that took, 2.251, seconds to produce.