A238750 Number T(n,k) of standard Young tableaux with n cells and largest value n in row k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 10, 7, 5, 3, 1, 0, 26, 20, 14, 11, 4, 1, 0, 76, 56, 44, 31, 19, 5, 1, 0, 232, 182, 139, 106, 69, 29, 6, 1, 0, 764, 589, 475, 351, 265, 127, 41, 7, 1, 0, 2620, 2088, 1658, 1303, 971, 583, 209, 55, 8, 1

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Mar 04 2014

Keywords

• nonn
• tabl

Comments

Also the number of ballot sequences of length n having last value k.

Also the number of standard Young tableaux with n cells where the row containing the largest value n has length k.

Also the number of ballot sequences of length n where the last value has multiplicity k.

T(0,0) = 1 by convention.

Columns k=0-2 give: A000007, A000085(n-1), A238124(n-1).

T(2n,n) gives A246731.

Row sums give A000085.

Examples

			The 10 tableaux with n=4 cells sorted by the number of the row containing the largest value 4 are:
:[1 4] [1 2 4] [1 3 4] [1 2 3 4]:[1 2] [1 3] [1 2 3]:[1 2] [1 3]:[1]:
:[2]   [3]     [2]              :[3 4] [2 4] [4]    :[3]   [2]  :[2]:
:[3]                            :                   :[4]   [4]  :[3]:
:                               :                   :           :[4]:
: --------------1-------------- : --------2-------- : ----3---- : 4 :
Their corresponding ballot sequences are: [1,2,3,1], [1,1,2,1], [1,2,1,1], [1,1,1,1], [1,1,2,2], [1,2,1,2], [1,1,1,2], [1,1,2,3], [1,2,1,3], [1,2,3,4].  Thus row 4 = [0, 4, 3, 2, 1].
Triangle T(n,k) begins:
00:   1;
01:   0,    1;
02:   0,    1,    1;
03:   0,    2,    1,    1;
04:   0,    4,    3,    2,    1;
05:   0,   10,    7,    5,    3,   1;
06:   0,   26,   20,   14,   11,   4,   1;
07:   0,   76,   56,   44,   31,  19,   5,   1;
08:   0,  232,  182,  139,  106,  69,  29,   6,  1;
09:   0,  764,  589,  475,  351, 265, 127,  41,  7,  1;
10:   0, 2620, 2088, 1658, 1303, 971, 583, 209, 55,  8,  1;
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened
• Wikipedia, Young tableau

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= proc(l) local n; n:=nops(l); `if`(n=0, 1, add(
     `if`(i=n or l[i]>l[i+1], x^i *h(subsop(i=
     `if`(i=n and l[n]=1, NULL, l[i]-1), l)), 0), i=1..n))
    end:
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]),
     add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, n, [])):
seq(T(n), n=0..12);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+
     Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[l_] := With[{ n = Length[l]}, If[n == 0, 1, Sum[
     If[i == n || l[[i]] > l[[i + 1]], x^i *h[ReplacePart[l, i ->
     If[i == n && l[[n]] == 1, Nothing, l[[i]] - 1]]], 0], {i, n}]]];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]],
     Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[b[n, n, {}], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code *)