A219272 - OEIS

%I #36 Feb 09 2017 09:17:27

%S 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,

%T 10,30,63,162,372,1506,3300,7887,15015,48048,292864,1,1,1,3,4,10,31,

%U 69,182,525,1911,5115,17347,43758,149721,626769,1946516,4934930

%N 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.

%C 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.

%H Alois P. Heinz, <a href="/A219272/b219272.txt">Columns k = 0..22, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

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

%e A(3,2) = 2:

%e +------+  +------+

%e | 1  2 |  | 1  3 |

%e | 3 .--+  | 2 .--+

%e +---+     +---+

%e A(3,3) = 3:

%e +------+  +------+  +---------+

%e | 1  2 |  | 1  3 |  | 1  2  3 |

%e | 3 .--+  | 2 .--+  +---------+

%e +---+     +---+

%e Triangle A(n,k) begins:

%e 1,  1,  1,  1,   1,    1,    1,    1,    1, ...

%e .   1,  1,  1,   1,    1,    1,    1,    1, ...

%e .       1,  1,   1,    1,    1,    1,    1, ...

%e .       2,  3,   3,    3,    3,    3,    3, ...

%e .           3,   4,    4,    4,    4,    4, ...

%e .           5,   9,   10,   10,   10,   10, ...

%e .          16,  25,   30,   31,   31,   31, ...

%e .               49,   63,   69,   70,   70, ...

%e .               70,  162,  182,  189,  190, ...

%p h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+

%p       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)

%p     end:

%p g:= proc(n, i, l) local s; s:=i*(i+1)/2;

%p       `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,

%p        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))

%p     end:

%p A:= (n, k)-> g(n, k, []):

%p seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);

%t 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}]];

%t 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]]]]]];

%t A[n_, k_] := g[n, k, {}];

%t 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_ *)

%Y Column heights are A000124.

%Y Column sums give: A219273.

%Y Diagonal gives: A218293.

%Y Leftmost nonzero elements give A219339.

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

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

%K nonn,tabf

%O 0,7

%A _Alois P. Heinz_, Nov 17 2012