cp's OEIS Frontend

A219339 Number of standard Young tableaux for partitions of n into distinct parts with largest part floor(sqrt(2*n)+1/2).

%I A219339 #26 Feb 17 2017 02:33:39
%S A219339 1,1,1,2,3,5,16,49,70,168,768,3300,7887,15015,48048,292864,1946516,
%T A219339 4934930,14454726,34918884,141892608,1100742656,9732668946,
%U A219339 32773404950,97848532782,344699731090,1020872973120,5091106775040,48608795688960,586393249199550
%N A219339 Number of standard Young tableaux for partitions of n into distinct parts with largest part floor(sqrt(2*n)+1/2).
%C A219339 a(n) is the leftmost nonzero element in row n of A219272, A219274.
%C A219339 Floor(sqrt(2*n)+1/2) = A002024(n) for n>0.  There are no partitions of n into distinct parts with a smaller largest part.
%H A219339 Alois P. Heinz, <a href="/A219339/b219339.txt">Table of n, a(n) for n = 0..300</a>
%H A219339 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A219339 a(n) = A219272(n,floor(sqrt(2*n)+1/2)) = A219274(n,floor(sqrt(2*n)+1/2)).
%e A219339 For n=5, we have floor(sqrt(2*n)+1/2) = 3, and a(5) = 5, because there are 5 standard Young tableaux for partitions of 5 into distinct parts with largest part 3:
%e A219339 +---------+  +---------+  +---------+  +---------+  +---------+
%e A219339 | 1  2  3 |  | 1  2  4 |  | 1  2  5 |  | 1  3  4 |  | 1  3  5 |
%e A219339 | 4  5 .--+  | 3  5 .--+  | 3  4 .--+  | 2  5 .--+  | 2  4 .--+
%e A219339 +------+     +------+     +------+     +------+     +------+
%p A219339 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A219339       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A219339     end:
%p A219339 g:= proc(n, i, l) local s; s:=i*(i+1)/2;
%p A219339       `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p A219339        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p A219339     end:
%p A219339 a:= n-> g(n, floor(sqrt(2*n)+1/2), []):
%p A219339 seq(a(n), n=0..30);
%t A219339 h[l_] := (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_] := g[n, i, l] = (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, Floor[Sqrt[2*n]+1/2], {}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 16 2017, translated from Maple *)
%Y A219339 Cf. A005118 (subsequence), A219347.
%K A219339 nonn
%O A219339 0,4
%A A219339 _Alois P. Heinz_, Nov 18 2012