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A214088 Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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 A276516 A253638

Adjacent sequences:  A214085 A214086 A214087 * A214089 A214090 A214091

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 02 2012

STATUS

approved

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Last modified June 23 11:24 EDT 2021. Contains 345397 sequences. (Running on oeis4.)