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%I #28 Feb 18 2015 05:19:40
%S 1,1,1,3,4,10,31,70,190,561,2191,6226,22683,74152,283349,1211354,
%T 4572672,18844177,77585825,327472752,1418056071,7083303437,
%U 31251988918,153456264178,723293387594,3596567095155,17360616601051,89955643932801,486526881887485,2551613423040841,14029592127656040,76756835252971657,428044848852530252
%N Number of standard Young tableaux with shapes corresponding to partitions into distinct parts.
%H Alois P. Heinz, <a href="/A218293/b218293.txt">Table of n, a(n) for n = 0..120</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%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, l) local s; s:=i*(i+1)/2;
%p `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p end:
%p a:= n-> g(n, n, []):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Nov 08 2012
%t h[l_List] := 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}]]; g[n_, i_, l_List] := 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]]]]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%Y Cf. A000085 (standard Young tableaux for all shapes).
%Y Diagonal of A219272, row sums of A219274, A219311. - _Alois P. Heinz_, Nov 17 2012
%Y Cf. A225121 (tableaux with shapes corresponding to partitions into distinct parts with minimal difference 2).
%K nonn
%O 0,4
%A _Joerg Arndt_, Oct 25 2012
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