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

0,6

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.

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

Formula

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

Example

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].

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 *)

Crossrefs

Column k=3 of A214088.

Sequence in context: A267365 A034536 A283538 * A179597 A295209 A151496

Adjacent sequences: A214456 A214457 A214458 * A214460 A214461 A214462

Keyword

nonn

Author

Alois P. Heinz, Jul 18 2012

Status

approved