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A219275 Number of standard Young tableaux for partitions of nonnegative integers into distinct parts with largest part n.

%I A219275 #19 Nov 02 2022 14:45:12
%S A219275 1,1,3,25,1069,368168,1299366501,55208013380403,32401197537296758130,
%T A219275 297072961835477978342245712,47538199827835784548062928051198402,
%U A219275 146779873623344672821145371965795071455181183,9581411392319396646028223743176779937161862866453789852
%N A219275 Number of standard Young tableaux for partitions of nonnegative integers into distinct parts with largest part n.
%H A219275 Alois P. Heinz, <a href="/A219275/b219275.txt">Table of n, a(n) for n = 0..22</a>
%H A219275 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A219275 a(2) = 3:
%e A219275 +------+  +------+  +------+
%e A219275 | 1  2 |  | 1  3 |  | 1  2 |
%e A219275 | 3 .--+  | 2 .--+  +------+
%e A219275 +---+     +---+
%p A219275 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219275       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219275     end:
%p A219275 b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):
%p A219275 a:= n-> `if`(n=0, 1, b(n-1, [n])):
%p A219275 seq(a(n), n=0..12);
%t A219275 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 A219275 b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];
%t A219275 a[n_] := If[n == 0, 1, b[n - 1, {n}]];
%t A219275 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 02 2022, after _Alois P. Heinz_ *)
%Y A219275 Column sums of A219274.
%Y A219275 First differences of A219273.
%K A219275 nonn
%O A219275 0,3
%A A219275 _Alois P. Heinz_, Nov 17 2012