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 A238794 Number T(n,k) of standard Young tableaux with n cells and k as last value in the first row; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 10
 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 4, 0, 1, 3, 5, 7, 10, 0, 1, 6, 10, 14, 19, 26, 0, 1, 10, 19, 29, 41, 56, 76, 0, 1, 20, 41, 66, 96, 132, 176, 232, 0, 1, 35, 86, 152, 232, 327, 441, 582, 764, 0, 1, 70, 197, 374, 596, 863, 1181, 1563, 2031, 2620 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS T(0,0) = 1 by convention. Also the number of ballot sequences of length n where k is the position of the last occurrence of the minimal value. Diagonal: T(n,n) = A000085(n-1) for n>=1. Columns k=0-10 give: A000007, A000012 for n>0, A001405(n-2) for n>1, A245001, A245002, A245003, A245004, A245005, A245006, A245007, A245008. T(2n,n) gives A245000. Row sums give A000085. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened Wikipedia, Young tableau EXAMPLE The 10 tableaux with n=4 cells sorted by last value in the first row: :[1]:[1 2] [1 2]:[1 3] [1 3] [1 2 3]:[1 4] [1 2 4] [1 3 4] [1 2 3 4]: :[2]:[3]   [3 4]:[2]   [2 4] [4]    :[2]   [3]     [2]              : :[3]:[4]        :[4]                :[3]                            : :[4]:           :                   :                               : : 1 : ----2---- : --------3-------- : --------------4-------------- : Their corresponding ballot sequences are: [1,2,3,4], [1,1,2,3], [1,1,2,2], [1,2,1,3], [1,2,1,2], [1,1,1,2], [1,2,3,1], [1,1,2,1], [1,2,1,1], [1,1,1,1].  Thus row 4 = [0, 1, 2, 3, 4]. Triangle T(n,k) begins: 00:   1; 01:   0, 1; 02:   0, 1,  1; 03:   0, 1,  1,   2; 04:   0, 1,  2,   3,   4; 05:   0, 1,  3,   5,   7,  10; 06:   0, 1,  6,  10,  14,  19,  26; 07:   0, 1, 10,  19,  29,  41,  56,   76; 08:   0, 1, 20,  41,  66,  96, 132,  176,  232; 09:   0, 1, 35,  86, 152, 232, 327,  441,  582,  764; 10:   0, 1, 70, 197, 374, 596, 863, 1181, 1563, 2031, 2620; MAPLE b:= proc(n, l) option remember; `if`(n=0, 1, add(`if`(       i=1 or l[i-1]>l[i], b(n-1, subsop(i=l[i]+1, l)), 0),       i=1..nops(l)) +(p-> p+(x^(1+add(j, j=l))-1)*       coeff(p, x, 0))(b(n-1, [l[], 1])))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, [])): seq(T(n), n=0..12); MATHEMATICA b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + Function[{p}, p + (x^(1+Total[l])-1)*Coefficient[p, x, 0]][b[n-1, Append[l, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, {}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 13 2015, translated from Maple *) CROSSREFS Sequence in context: A255120 A218601 A262678 * A135317 A332663 A227179 Adjacent sequences:  A238791 A238792 A238793 * A238795 A238796 A238797 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Mar 05 2014 STATUS approved

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Last modified January 16 09:13 EST 2021. Contains 340204 sequences. (Running on oeis4.)