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

0,34

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

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

Example

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, ...

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 *)

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.

Sequence in context: A085975 A277778 A255319 * A005091 A353455 A276516

Adjacent sequences: A214085 A214086 A214087 * A214089 A214090 A214091

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 02 2012

Status

approved