A219311 Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 9, 0, 1, 14, 16, 0, 1, 34, 35, 0, 1, 55, 134, 0, 1, 125, 435, 0, 1, 209, 1213, 768, 0, 1, 461, 3454, 2310, 0, 1, 791, 10484, 11407, 0, 1, 1715, 28249, 44187, 0, 1, 3002, 80302, 200044, 0, 1, 6434, 231895, 680160, 292864

Offset

0,8

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=A003056(n). T(n,k) = 0 for k>A003056(n).

Links

Alois P. Heinz, Rows n = 0..100, flattened

Wikipedia, Young tableau

Example

A(4,2) = 3:
  +---------+  +---------+  +---------+
  | 1  2  3 |  | 1  2  4 |  | 1  3  4 |
  | 4 .-----+  | 3 .-----+  | 2 .-----+
  +---+        +---+        +---+
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1;
  0,  1,    2;
  0,  1,    3;
  0,  1,    9;
  0,  1,   14,    16;
  0,  1,   34,    35;
  0,  1,   55,   134;
  0,  1,  125,   435;
  0,  1,  209,  1213,   768;
  0,  1,  461,  3454,  2310;
  0,  1,  791, 10484, 11407;
  ...

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(n, i, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0,
      g(n, i-1, min(k, i-1), l)+`if`(i>n, 0, g(n-i, i-1, k-1, [l[], i]))))
    end:
A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ];
g[n_, i_, k_, l_] := If[n == 0, h[l], If[n > k*(i-(k-1)/2), 0, g[n, i-1, Min[k, i-1], l] + If[i > n, 0, g[n-i, i-1, k-1, Append[l, i]]]]];
a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]];
t[n_, k_] := a[n, k] - a[n, k-1];
Table[Table[t[n, k], {k, 0, Floor[(Sqrt[1+8*n]-1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

Crossrefs

Columns k=0-10 give: A000007, A000012 (for n>0), A047171(n) = A037952(n)-1, A219316, A219317, A219318, A219319, A219320, A219321, A219322, A219323.

Row sums give: A218293.

Row lengths are 1 + A003056(n).

T(A000217(k),k) = A005118(k+1).

Cf. A219272, A219274.

Sequence in context: A029297 A289871 A257563 * A363393 A022880 A366614

Adjacent sequences: A219308 A219309 A219310 * A219312 A219313 A219314

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 17 2012

Status

approved