%I #30 Sep 08 2021 06:50:57

%S 1,0,1,0,1,0,1,2,0,1,3,0,1,9,0,1,14,16,0,1,34,35,0,1,55,134,0,1,125,

%T 435,0,1,209,1213,768,0,1,461,3454,2310,0,1,791,10484,11407,0,1,1715,

%U 28249,44187,0,1,3002,80302,200044,0,1,6434,231895,680160,292864

%N Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

%C T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=A003056(n). T(n,k) = 0 for k>A003056(n).

%H Alois P. Heinz, <a href="/A219311/b219311.txt">Rows n = 0..100, flattened</a>

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

%e A(4,2) = 3:

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

%e | 1 2 3 | | 1 2 4 | | 1 3 4 |

%e | 4 .-----+ | 3 .-----+ | 2 .-----+

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

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 1, 3;

%e 0, 1, 9;

%e 0, 1, 14, 16;

%e 0, 1, 34, 35;

%e 0, 1, 55, 134;

%e 0, 1, 125, 435;

%e 0, 1, 209, 1213, 768;

%e 0, 1, 461, 3454, 2310;

%e 0, 1, 791, 10484, 11407;

%e ...

%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, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0,

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

%p end:

%p A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end:

%p T:= (n, k)-> A(n, k) -A(n, k-1):

%p seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);

%t h[l_] := With[{n = Length[l]}, Sum[i, {i, 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_, k_, l_] := If[n == 0, h[l], If[n > k*(i-(k-1)/2), 0, g[n, i-1, Min[k, i-1], l] + If[i > n, 0, g[n-i, i-1, k-1, Append[l, i]]]]];

%t a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]];

%t t[n_, k_] := a[n, k] - a[n, k-1];

%t Table[Table[t[n, k], {k, 0, Floor[(Sqrt[1+8*n]-1)/2]}], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Dec 17 2013, translated from Maple *)

%Y Columns k=0-10 give: A000007, A000012 (for n>0), A047171(n) = A037952(n)-1, A219316, A219317, A219318, A219319, A219320, A219321, A219322, A219323.

%Y Row sums give: A218293.

%Y Row lengths are 1 + A003056(n).

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

%Y Cf. A219272, A219274.

%K nonn,tabf

%O 0,8

%A _Alois P. Heinz_, Nov 17 2012