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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.
7
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
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 17 2012
STATUS
approved