A214459 Number of n X 3 nonconsecutive chess tableaux.

1, 0, 0, 1, 1, 7, 27, 128, 640, 3351, 18313, 103404, 600538, 3571717, 21683185, 134005373, 841259885, 5355078350, 34512405410, 224908338137, 1480420941781, 9833512593113, 65860442383487, 444453988418791, 3020274890688447, 20656019108074552, 142107550142684602

Offset: 0

Author

Alois P. Heinz, Jul 18 2012

Keywords

• nonn

Comments

A standard Young tableau (SYT) with cell(i,j)+i+j == 1 mod 2 for all cells where entries m and m+1 never appear in the same row is called a nonconsecutive chess tableau.

Examples

			a(5) = 7:
  [1  6 11] [1  4 11] [1  6  9] [1  4  9] [1  4  7] [1  4  7] [1  4  7]
  [2  7 12] [2  5 12] [2  7 10] [2  5 10] [2  5 10] [2  5 10] [2  5  8]
  [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  6 13] [3 10 13]
  [4  9 14] [6  9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
  [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..70 from Alois P. Heinz)
• T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
• Jonas Sjöstrand, On the sign-imbalance of partition shapes, arXiv:math/0309231v3 [math.CO], 2005.
• Wikipedia, Young tableau

Crossrefs

Column k=3 of A214088.

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 irem(s+i-l[i], 2)=1 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..25);

Mathematica

b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, l}]}; If[s == 0, 1, Sum[If[t != i && Mod[s + i - l[[i]], 2] == 1 && l[[i]] > If[i == n, 0, l[[i + 1]]], b[ReplacePart[l, {i -> l[[i]] - 1}], i], 0], {i, 1, n}]]]; a[n_] := If[n < 1, 1, b[Array[3&, n], 0]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 13 2017, after Alois P. Heinz *)

Formula

a(n) ~ c * 8^n / n^4, where c = 0.250879571... - Vaclav Kotesovec, Sep 06 2017