A214461 Number of 5 X n nonconsecutive chess tableaux.

1, 1, 2, 7, 35, 212, 1421, 10128, 75724, 593905, 4927764, 43884083, 426665219, 4591883867, 54943702996, 725753304587, 10428313247758, 160361700765626, 2602492613187389, 44111609147837212, 775162479337156853, 14049351313802046511, 261640973700411314373

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(3) = 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..114 (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

Row n=5 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([n$5], 0):
seq(a(n), n=0..25);

Mathematica

b[l_List, t_] := b[l, t] = With[{n = Length[l], s = Total[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_] := b[Table[n, {5}], 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, translated from Maple *)