A238727 Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 1, 2, 7, 0, 0, 1, 3, 8, 14, 0, 0, 1, 4, 11, 19, 41, 0, 0, 1, 7, 19, 34, 64, 107, 0, 0, 1, 11, 32, 62, 119, 202, 337, 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066, 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Mar 03 2014

Keywords

• nonn
• tabl

Comments

T(0,0) = 1 by convention.

Also the number of ballot sequences of length n having the last occurrence of the maximal value at position k.

T(n,3) = A051920(n-3) for n>3.

T(2*n,n) gives A246818.

Main diagonal gives A238728.

Row sums give A000085.

Examples

			The 10 tableaux with n=4 cells sorted by largest value in the last row:
  :[1 3 4]:[1 4] [1 2 4]:[1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4]:
  :[2]    :[2]   [3]    :[2] [3]   [2]   [4]     [3 4] [2 4]          :
  :       :[3]          :[3] [4]   [4]                                :
  :       :             :[4]                                          :
  : --2-- : -----3----- : ---------------------4--------------------- :
The 10 ballot sequences of length 4 sorted by the position of the last occurrence of the maximal value:
  [1, 2, 1, 1]  ->  2 } -- 1
  [1, 2, 3, 1]  ->  3 \ __ 2
  [1, 1, 2, 1]  ->  3 /
  [1, 2, 3, 4]  ->  4 \
  [1, 1, 2, 3]  ->  4  \
  [1, 2, 1, 3]  ->  4   \
  [1, 1, 1, 2]  ->  4    } 7
  [1, 1, 2, 2]  ->  4   /
  [1, 2, 1, 2]  ->  4  /
  [1, 1, 1, 1]  ->  4 /
thus row 4 = [0, 0, 1, 2, 7].
Triangle T(n,k) begins:
  00:   1;
  01:   0, 1;
  02:   0, 0, 2;
  03:   0, 0, 1,  3;
  04:   0, 0, 1,  2,   7;
  05:   0, 0, 1,  3,   8,  14;
  06:   0, 0, 1,  4,  11,  19,  41;
  07:   0, 0, 1,  7,  19,  34,  64, 107;
  08:   0, 0, 1, 11,  32,  62, 119, 202,  337;
  09:   0, 0, 1, 21,  64, 131, 248, 418,  671, 1066;
  10:   0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691;
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..43, flattened
• Wikipedia, Young tableau

Programs

Maple

h:= proc(l) option remember; local n, s; n:= nops(l); s:= add(i, i=l);
     `if`(n=0, 1, add(`if`(il[i+1], h(subsop(i=l[i]-1, l)),
     `if`(i=n, (p->add(coeff(p,x,j)*x^`if`(j1, l[i]-1, [][]), l))), 0)), i=1..n))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]),
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p->seq(coeff(p, x, i), i=0..n))(g(n$2, [])):
seq(T(n), n=0..12);

Mathematica

h[l_] := h[l] = With[{n = Length[l], s = Total[l]},
     If[n == 0, 1, Sum[If[i < n && l[[i]] > l[[i + 1]],
     h[ReplacePart[l, i -> l[[i]] - 1]], If[i == n, Function[p,
     Sum[Coefficient[p, x, j] x^If[j < s, s, j], {j, 0,
     Exponent[p, x]}]][h[ReplacePart[l, i -> If[l[[i]] > 1,
     l[[i]] - 1, Nothing]]]], 0]], {i, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]],
     Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[g[n, n, {}], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code *)