A219274 Number T(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

1, 1, 1, 2, 1, 3, 5, 16, 1, 4, 9, 49, 70, 168, 768, 1, 5, 14, 92, 204, 738, 3300, 7887, 15015, 48048, 292864, 1, 6, 20, 153, 405, 1815, 9460, 28743, 101673, 333905, 1946516, 4934930, 14454726, 34918884, 141892608, 1100742656, 1, 7, 27, 235, 715, 3630, 21307

Offset

0,4

Comments

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

Links

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

Wikipedia, Young tableau

Formula

T(n,k) = A219272(n,k) - A219272(n,k-1) for k>0.

Example

T(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
Triangle T(n,k) begins:
1;
.  1;
.     1;
.     2,  1;
.         3,   1;
.         5,   4,   1;
.        16,   9,   5,   1;
.             49,  14,   6,   1;
.             70,  92,  20,   7,  1;
.            168, 204, 153,  27,  8, 1;
.            768, 738, 405, 235, 35, 9, 1;

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:
T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);

Mathematica

h[l_] := Module[{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_] := Module[{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]]]]]];
T[n_, k_] := If[k > n, 0, g[n - k, k - 1, {k}]];
Table[Table[T[n, k], {n, k, k(k + 1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)

Crossrefs

Column heights are A000124(k-1) for k>0.

Column sums give: A219275.

Row sums give: A218293.

Diagonal and lower diagonals give: A000012, A000027 (for n>1), A000096(n-1) (for n>2).

Leftmost nonzero elements give A219339.

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

Triangle read by rows reversed gives: A219356.

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

Sequence in context: A096631 A144057 A272891 * A241498 A143581 A096871

Adjacent sequences: A219271 A219272 A219273 * A219275 A219276 A219277

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 17 2012

Status

approved