A219275 - OEIS

%I #19 Nov 02 2022 14:45:12

%S 1,1,3,25,1069,368168,1299366501,55208013380403,32401197537296758130,

%T 297072961835477978342245712,47538199827835784548062928051198402,

%U 146779873623344672821145371965795071455181183,9581411392319396646028223743176779937161862866453789852

%N Number of standard Young tableaux for partitions of nonnegative integers into distinct parts with largest part n.

%H Alois P. Heinz, <a href="/A219275/b219275.txt">Table of n, a(n) for n = 0..22</a>

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

%e a(2) = 3:

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

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

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

%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 b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):

%p a:= n-> `if`(n=0, 1, b(n-1, [n])):

%p seq(a(n), n=0..12);

%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 b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];

%t a[n_] := If[n == 0, 1, b[n - 1, {n}]];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 02 2022, after _Alois P. Heinz_ *)

%Y Column sums of A219274.

%Y First differences of A219273.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 17 2012