login
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.
8
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
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 17 2012
STATUS
approved