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

1, 1, 1, 2, 3, 5, 16, 49, 70, 168, 768, 3300, 7887, 15015, 48048, 292864, 1946516, 4934930, 14454726, 34918884, 141892608, 1100742656, 9732668946, 32773404950, 97848532782, 344699731090, 1020872973120, 5091106775040, 48608795688960, 586393249199550

Offset: 0

Author

Alois P. Heinz, Nov 18 2012

Keywords

• nonn

Comments

a(n) is the leftmost nonzero element in row n of A219272, A219274.

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.

Examples

			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:
+---------+  +---------+  +---------+  +---------+  +---------+
| 1  2  3 |  | 1  2  4 |  | 1  2  5 |  | 1  3  4 |  | 1  3  5 |
| 4  5 .--+  | 3  5 .--+  | 3  4 .--+  | 2  5 .--+  | 2  4 .--+
+------+     +------+     +------+     +------+     +------+
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Young tableau

Crossrefs

Cf. A005118 (subsequence), A219347.

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
a:= n-> g(n, floor(sqrt(2*n)+1/2), []):
seq(a(n), n=0..30);

Mathematica

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 *)

Formula

a(n) = A219272(n,floor(sqrt(2*n)+1/2)) = A219274(n,floor(sqrt(2*n)+1/2)).