OFFSET
0,25
COMMENTS
A standard Young tableau (SYT) with cell(i,j)+i+j == 1 mod 2 for all cells is called a chess tableau. The definition appears first in the article by Jonas Sjöstrand.
LINKS
Alois P. Heinz, Antidiagonals n = 0..24, 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(4,3) = A(3,4) = 6:
[1 4 7] [1 4 5] [1 2 3] [1 4 7] [ 1 4 7] [ 1 2 3]
[2 5 10] [2 7 10] [4 7 10] [2 5 10] [ 2 5 8] [ 4 5 6]
[3 8 11] [3 8 11] [5 8 11] [3 6 11] [ 3 6 9] [ 7 8 9]
[6 9 12] [6 9 12] [6 9 12] [8 9 12] [10 11 12] [10 11 12].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 0, 1, 0, 2, 0, 5, ...
1, 1, 1, 2, 6, 22, 92, 422, ...
1, 1, 0, 6, 0, 324, 0, 47570, ...
1, 1, 2, 22, 324, 8716, 343234, 17423496, ...
1, 1, 0, 92, 0, 343234, 0, 8364334408, ...
1, 1, 5, 422, 47570, 17423496, 8364334408, 6873642982160, ...
MAPLE
b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0,
args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs))
end:
A:= (n, k)-> `if`(n<k, A(k, n), `if`(k<2, 1, b(n$k))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[args_List] := b[args] = Module[{s = Total[args], nargs = Length[args]}, If[s == 0, 1, Sum[If[Mod[s + i - args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i + 1]]], b[ReplacePart[args, i -> args[[i]] - 1]], 0], {i, 1, nargs}]]]; A[n_, k_] := If[n < k, A[k, n], If[k < 2, 1, b[Array[n &, k]]]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 01 2012
STATUS
approved