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A219311 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.

%I A219311 #30 Sep 08 2021 06:50:57
%S A219311 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 A219311 435,0,1,209,1213,768,0,1,461,3454,2310,0,1,791,10484,11407,0,1,1715,
%U A219311 28249,44187,0,1,3002,80302,200044,0,1,6434,231895,680160,292864
%N A219311 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 A219311 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 A219311 Alois P. Heinz, <a href="/A219311/b219311.txt">Rows n = 0..100, flattened</a>
%H A219311 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A219311 A(4,2) = 3:
%e A219311   +---------+  +---------+  +---------+
%e A219311   | 1  2  3 |  | 1  2  4 |  | 1  3  4 |
%e A219311   | 4 .-----+  | 3 .-----+  | 2 .-----+
%e A219311   +---+        +---+        +---+
%e A219311 Triangle T(n,k) begins:
%e A219311   1;
%e A219311   0,  1;
%e A219311   0,  1;
%e A219311   0,  1,    2;
%e A219311   0,  1,    3;
%e A219311   0,  1,    9;
%e A219311   0,  1,   14,    16;
%e A219311   0,  1,   34,    35;
%e A219311   0,  1,   55,   134;
%e A219311   0,  1,  125,   435;
%e A219311   0,  1,  209,  1213,   768;
%e A219311   0,  1,  461,  3454,  2310;
%e A219311   0,  1,  791, 10484, 11407;
%e A219311   ...
%p A219311 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219311       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219311     end:
%p A219311 g:= proc(n, i, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0,
%p A219311       g(n, i-1, min(k, i-1), l)+`if`(i>n, 0, g(n-i, i-1, k-1, [l[], i]))))
%p A219311     end:
%p A219311 A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end:
%p A219311 T:= (n, k)-> A(n, k) -A(n, k-1):
%p A219311 seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);
%t A219311 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 A219311 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 A219311 a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]];
%t A219311 t[n_, k_] := a[n, k] - a[n, k-1];
%t A219311 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 A219311 Columns k=0-10 give: A000007, A000012 (for n>0), A047171(n) = A037952(n)-1, A219316, A219317, A219318, A219319, A219320, A219321, A219322, A219323.
%Y A219311 Row sums give: A218293.
%Y A219311 Row lengths are 1 + A003056(n).
%Y A219311 T(A000217(k),k) = A005118(k+1).
%Y A219311 Cf. A219272, A219274.
%K A219311 nonn,tabf
%O A219311 0,8
%A A219311 _Alois P. Heinz_, Nov 17 2012