A213932 Number of solid standard Young tableaux of shape [[n,n,n],[n,n]].

1, 5, 707, 268326, 168146839, 143163177336, 149998192424502, 182598353781240533, 249032962712552804432, 371285830572997665257695, 594729699502746726969433566, 1010574132470951359396337494800, 1804193873947216124589237862262262

Offset: 0

Author

Alois P. Heinz, Jul 23 2012

Keywords

• nonn

Comments

Also the number of solid standard Young tableaux of shape [[n,n],[n,n],[n]].

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..30
• S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
• Wikipedia, Young tableau

Crossrefs

Cf. A213978, A214637, A214638.

Programs

Maple

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n, n, n], [n, n]]):
seq(a(n), n=0..10);

Mathematica

b[l_] := b[l] = With[{ m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]]; a[n_] := b[{{n, n, n}, {n, n}}]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 18 2013, translated from Maple *)