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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.

%I A219272 #36 Feb 09 2017 09:17:27
%S A219272 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 A219272 10,30,63,162,372,1506,3300,7887,15015,48048,292864,1,1,1,3,4,10,31,
%U A219272 69,182,525,1911,5115,17347,43758,149721,626769,1946516,4934930
%N 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.
%C A219272 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 A219272 Alois P. Heinz, <a href="/A219272/b219272.txt">Columns k = 0..22, flattened</a>
%H A219272 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A219272 T(n,k) = Sum_{i=0..k} A219274(n,i).
%e A219272 A(3,2) = 2:
%e A219272 +------+  +------+
%e A219272 | 1  2 |  | 1  3 |
%e A219272 | 3 .--+  | 2 .--+
%e A219272 +---+     +---+
%e A219272 A(3,3) = 3:
%e A219272 +------+  +------+  +---------+
%e A219272 | 1  2 |  | 1  3 |  | 1  2  3 |
%e A219272 | 3 .--+  | 2 .--+  +---------+
%e A219272 +---+     +---+
%e A219272 Triangle A(n,k) begins:
%e A219272 1,  1,  1,  1,   1,    1,    1,    1,    1, ...
%e A219272 .   1,  1,  1,   1,    1,    1,    1,    1, ...
%e A219272 .       1,  1,   1,    1,    1,    1,    1, ...
%e A219272 .       2,  3,   3,    3,    3,    3,    3, ...
%e A219272 .           3,   4,    4,    4,    4,    4, ...
%e A219272 .           5,   9,   10,   10,   10,   10, ...
%e A219272 .          16,  25,   30,   31,   31,   31, ...
%e A219272 .               49,   63,   69,   70,   70, ...
%e A219272 .               70,  162,  182,  189,  190, ...
%p A219272 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219272       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219272     end:
%p A219272 g:= proc(n, i, l) local s; s:=i*(i+1)/2;
%p A219272       `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p A219272        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p A219272     end:
%p A219272 A:= (n, k)-> g(n, k, []):
%p A219272 seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);
%t A219272 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 A219272 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 A219272 A[n_, k_] := g[n, k, {}];
%t A219272 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 A219272 Column heights are A000124.
%Y A219272 Column sums give: A219273.
%Y A219272 Diagonal gives: A218293.
%Y A219272 Leftmost nonzero elements give A219339.
%Y A219272 Column of leftmost nonzero element is A002024(n) for n>0.
%Y A219272 T(A000217(n),n) = A005118(n+1).
%K A219272 nonn,tabf
%O A219272 0,7
%A A219272 _Alois P. Heinz_, Nov 17 2012