A219272 Number A(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part <= k; triangle A(n,k), k>=0, 0<=n<=k*(k+1)/2, read by columns.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 5, 16, 1, 1, 1, 3, 4, 9, 25, 49, 70, 168, 768, 1, 1, 1, 3, 4, 10, 30, 63, 162, 372, 1506, 3300, 7887, 15015, 48048, 292864, 1, 1, 1, 3, 4, 10, 31, 69, 182, 525, 1911, 5115, 17347, 43758, 149721, 626769, 1946516, 4934930

Offset

0,7

Comments

A(n,k) is defined for n,k >= 0. A(n,k) = 0 iff n > k*(k+1)/2 = A000217(k). The triangle contains only the nonzero terms. A(n,k) = A(n,n) for k>=n.

Links

Alois P. Heinz, Columns k = 0..22, flattened

Wikipedia, Young tableau

Formula

T(n,k) = Sum_{i=0..k} A219274(n,i).

Example

A(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
A(3,3) = 3:
+------+  +------+  +---------+
| 1  2 |  | 1  3 |  | 1  2  3 |
| 3 .--+  | 2 .--+  +---------+
+---+     +---+
Triangle A(n,k) begins:
1,  1,  1,  1,   1,    1,    1,    1,    1, ...
.   1,  1,  1,   1,    1,    1,    1,    1, ...
.       1,  1,   1,    1,    1,    1,    1, ...
.       2,  3,   3,    3,    3,    3,    3, ...
.           3,   4,    4,    4,    4,    4, ...
.           5,   9,   10,   10,   10,   10, ...
.          16,  25,   30,   31,   31,   31, ...
.               49,   63,   69,   70,   70, ...
.               70,  162,  182,  189,  190, ...

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, l) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);

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, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = With[{s=i*(i+1)/2}, If[n==s, h[Join[l, Table[ i-j, {j, 0, i-1}]]], If[n>s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Append[l, i]]]]]];
A[n_, k_] := g[n, k, {}];
Table[Table[A[n, k], {n, 0, k*(k+1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

Crossrefs

Column heights are A000124.

Column sums give: A219273.

Diagonal gives: A218293.

Leftmost nonzero elements give A219339.

Column of leftmost nonzero element is A002024(n) for n>0.

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

Sequence in context: A307689 A299037 A173072 * A084097 A306684 A293991

Adjacent sequences: A219269 A219270 A219271 * A219273 A219274 A219275

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 17 2012

Status

approved