A238728 Number of standard Young tableaux with n cells where the largest value n is contained in the last row.

1, 1, 2, 3, 7, 14, 41, 107, 337, 1066, 3691, 12962, 49061, 188894, 766845, 3182844, 13758383, 60858842, 278312475, 1301323108, 6258671365, 30742575588, 154785692507, 794888735945, 4173162573277, 22318859784416, 121767607626621, 676010926754742

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Mar 03 2014

Keywords

• nonn

Comments

a(0) = 1 by convention.

Also number of ballot sequences of length n where the last position has a maximal value.

Main diagonal of A238727.

Examples

			a(4) = 7 counts the following tableaux:
  [1]  [1 2]  [1 3]  [1 2 3]  [1 2]  [1 3]  [1 2 3 4]
  [2]  [3]    [2]    [4]      [3 4]  [2 4]
  [3]  [4]    [4]
  [4]
corresponding to the following ballot sequences: [1,2,3,4], [1,1,2,3], [1,2,1,3], [1,1,1,2], [1,1,2,2], [1,2,1,2], [1,1,1,1].
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..60
• Wikipedia, Young tableau

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:= l->`if`(l=[], 1, h(subsop(-1=`if`(l[-1]=1, [][], l[-1]-1), l))):
b:= proc(n, i, l) `if`(n=0 or i=1, g([l[], 1$n]),
      add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)) end:
a:= n-> b(n, n, []):
seq(a(n), n=0..28);

Mathematica

h[l_List] := 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}]]; g[l_List] := If[l == {}, 1, h[If[Last[l] == 1, Most[l], Append[Most[l], Last[l]-1]]]]; b[n_, i_, l_List] := If[n == 0 || i == 1, g[Join[l, Array[1&, n]]], Sum[b[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Feb 12 2015, after Maple *)