A214460 Number of 4 X 2*n+1 nonconsecutive chess tableaux.

1, 1, 1, 1, 2, 9, 44, 233, 1472, 10610, 82828, 688511, 6042580, 55523953, 530498426, 5242548633, 53361716373, 557495995958, 5961086843092, 65077096683358, 723862991146939, 8189257485453446, 94087399914391254, 1096331953010025684, 12941148147430923798

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(4) = 2:
  [1, 4,  7, 12, 15, 20, 23, 28, 31]  [1,   4,  7, 10, 13, 16, 19, 22, 25]
  [2, 5, 10, 13, 18, 21, 26, 29, 34]  [2,   5,  8, 11, 14, 17, 28, 31, 34]
  [3, 8, 11, 16, 19, 24, 27, 32, 35]  [3,   6,  9, 20, 23, 26, 29, 32, 35]
  [6, 9, 14, 17, 22, 25, 30, 33, 36]  [12, 15, 18, 21, 24, 27, 30, 33, 36].
		

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

Bisection (odd part) of row n=4 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([(2*n+1)$4], 0):
seq(a(n), n=0..25);

Mathematica

b[l_List, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], 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[{2n+1, 2n+1, 2n+1, 2n+1}, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, translated from Maple *)

Formula

a(n) ~ c * 16^n / n^(15/2), where c = 5.347555... - Vaclav Kotesovec, Dec 05 2017