A214088 Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 7, 0, 1, 1, 1, 0, 0, 1, 1, 35, 27, 5, 1, 1, 1, 0, 0, 1, 0, 212, 0, 128, 0, 1, 1, 1, 0, 0, 1, 1, 1421, 5075, 6212, 640, 14, 1, 1, 1, 0, 0, 1, 0, 10128, 0, 430275, 0, 3351, 0, 1, 1

Offset: 0

Author

Alois P. Heinz, Jul 02 2012

Keywords

• nonn
• tabl

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(3,5) = 1:
  [1 4 7 10 13]
  [2 5 8 11 14]
  [3 6 9 12 15].
A(7,2) = 5:
  [1  8]   [1  6]   [1  4]   [1   6]   [1   4]
  [2  9]   [2  7]   [2  5]   [2   7]   [2   5]
  [3 10]   [3 10]   [3 10]   [3   8]   [3   8]
  [4 11]   [4 11]   [6 11]   [4   9]   [6   9]
  [5 12]   [5 12]   [7 12]   [5  12]   [7  12]
  [6 13]   [8 13]   [8 13]   [10 13]   [10 13]
  [7 14]   [9 14]   [9 14]   [11 14]   [11 14].
Square array A(n,k) begins:
  1,  1,  1,   1,    1,      1,        1,          1, ...
  1,  1,  0,   0,    0,      0,        0,          0, ...
  1,  1,  0,   0,    0,      0,        0,          0, ...
  1,  1,  1,   1,    1,      1,        1,          1, ...
  1,  1,  0,   1,    0,      1,        0,          1, ...
  1,  1,  2,   7,   35,    212,     1421,      10128, ...
  1,  1,  0,  27,    0,   5075,        0,    2402696, ...
  1,  1,  5, 128, 6212, 430275, 42563460, 5601745187, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..21, flattened
• 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

Cf. A000108 (bisection of column k=2 for n>0), A214459 (column k=3), A214460 (bisection of row n=4), A214461 (row n=5), A214020, 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 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, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)