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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-Fran├žois 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.)