A214159 Number of n X 3 nonconsecutive tableaux.

1, 0, 1, 6, 72, 960, 14257, 228738, 3896062, 69590192, 1291977768, 24767357824, 487787117424, 9830929279248, 202119964701201, 4228355224663002, 89820746194643154, 1934075860467190608, 42153272431384984926, 928785495190695948612, 20666589509182675178064

Offset: 0

Author

Alois P. Heinz, Jul 05 2012

Keywords

• nonn

Comments

A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.

Examples

			a(2) = 1:
  [1 3 5]
  [2 4 6].
a(3) = 6:
  [1 4 7]   [1 3 7]   [1 4 6]   [1 3 6]   [1 3 6]   [1 3 5]
  [2 5 8]   [2 5 8]   [2 5 8]   [2 5 8]   [2 4 8]   [2 6 8]
  [3 6 9]   [4 6 9]   [3 7 9]   [4 7 9]   [5 7 9]   [4 7 9].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700
• T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
• Wikipedia, Young tableau

Crossrefs

Column k=3 of A214021.

Programs

Maple

b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
       add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
    end:
a:= n-> b([3$n], 0):
seq(a(n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 0, 1][n+1],
      ((22120*n^5 -58226*n^4 +31547*n^3 +42158*n^2 -53589*n +16146)*a(n-1)
      -(5988 -27068*n +50360*n^2 -62154*n^3 +49630*n^4 -15400*n^5)*a(n-2)
      -(7560*n^5 -23058*n^4 +24333*n^3 -11976*n^2 +2967*n -306)*a(n-3))
      /(6*(n+2)*(140*n^2-357*n+234)*(n+1)^2))
    end:
seq(a(n), n=0..25);

Mathematica

b[l_, t_] := b[l, t] = Module[{n = Length[l], s = Total[l]}, If[s == 0, 1, Sum[If[t != i && l[[i]] > If[i == n, 0, l[[i + 1]]], b[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, n}]]];
a[n_] := b[Table[3, n], 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from first Maple program *)

Formula

a(n) ~ 3^(3*n+13/2) / (2^12 * Pi * n^4). - Vaclav Kotesovec, Jul 16 2014