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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.

%I A219274 #36 Sep 01 2023 08:51:04
%S A219274 1,1,1,2,1,3,5,16,1,4,9,49,70,168,768,1,5,14,92,204,738,3300,7887,
%T A219274 15015,48048,292864,1,6,20,153,405,1815,9460,28743,101673,333905,
%U A219274 1946516,4934930,14454726,34918884,141892608,1100742656,1,7,27,235,715,3630,21307
%N 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.
%C A219274 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.
%H A219274 Alois P. Heinz, <a href="/A219274/b219274.txt">Columns k = 0..22, flattened</a>
%H A219274 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A219274 T(n,k) = A219272(n,k) - A219272(n,k-1) for k>0.
%e A219274 T(3,2) = 2:
%e A219274 +------+  +------+
%e A219274 | 1  2 |  | 1  3 |
%e A219274 | 3 .--+  | 2 .--+
%e A219274 +---+     +---+
%e A219274 Triangle T(n,k) begins:
%e A219274 1;
%e A219274 .  1;
%e A219274 .     1;
%e A219274 .     2,  1;
%e A219274 .         3,   1;
%e A219274 .         5,   4,   1;
%e A219274 .        16,   9,   5,   1;
%e A219274 .             49,  14,   6,   1;
%e A219274 .             70,  92,  20,   7,  1;
%e A219274 .            168, 204, 153,  27,  8, 1;
%e A219274 .            768, 738, 405, 235, 35, 9, 1;
%p A219274 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219274       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219274     end:
%p A219274 g:= proc(n, i, l) local s; s:=i*(i+1)/2;
%p A219274       `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p A219274        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p A219274     end:
%p A219274 T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
%p A219274 seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);
%t A219274 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}]];
%t A219274 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 A219274 T[n_, k_] := If[k > n, 0, g[n - k, k - 1, {k}]];
%t A219274 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_ *)
%Y A219274 Column heights are A000124(k-1) for k>0.
%Y A219274 Column sums give: A219275.
%Y A219274 Row sums give: A218293.
%Y A219274 Diagonal and lower diagonals give: A000012, A000027 (for n>1), A000096(n-1) (for n>2).
%Y A219274 Leftmost nonzero elements give A219339.
%Y A219274 Column of leftmost nonzero element is A002024(n) for n>0.
%Y A219274 Triangle read by rows reversed gives: A219356.
%Y A219274 T(A000217(n),n) = A005118(n+1).
%K A219274 nonn,tabf
%O A219274 0,4
%A A219274 _Alois P. Heinz_, Nov 17 2012