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

0,3

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.

Links

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

Wikipedia, Young tableau

Example

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].

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 *)

Crossrefs

Sequence in context: A274538 A113182 A165433 * A271718 A222626 A006627

Adjacent sequences: A238725 A238726 A238727 * A238729 A238730 A238731

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Mar 03 2014

Status

approved