%I #31 Oct 05 2018 20:10:02
%S 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,
%T 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,
%U 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
%N Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C 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.
%H Alois P. Heinz, <a href="/A214088/b214088.txt">Antidiagonals n = 0..21, flattened</a>
%H T. Y. Chow, H. Eriksson and C. K. Fan, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2a3.html">Chess tableaux</a>, Elect. J. Combin., 11 (2) (2005), #A3.
%H Jonas Sjöstrand, <a href="https://arxiv.org/abs/math/0309231v3">On the sign-imbalance of partition shapes</a>, arXiv:math/0309231v3 [math.CO], 2005.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A(3,5) = 1:
%e [1 4 7 10 13]
%e [2 5 8 11 14]
%e [3 6 9 12 15].
%e A(7,2) = 5:
%e [1 8] [1 6] [1 4] [1 6] [1 4]
%e [2 9] [2 7] [2 5] [2 7] [2 5]
%e [3 10] [3 10] [3 10] [3 8] [3 8]
%e [4 11] [4 11] [6 11] [4 9] [6 9]
%e [5 12] [5 12] [7 12] [5 12] [7 12]
%e [6 13] [8 13] [8 13] [10 13] [10 13]
%e [7 14] [9 14] [9 14] [11 14] [11 14].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 0, 1, 0, 1, 0, 1, ...
%e 1, 1, 2, 7, 35, 212, 1421, 10128, ...
%e 1, 1, 0, 27, 0, 5075, 0, 2402696, ...
%e 1, 1, 5, 128, 6212, 430275, 42563460, 5601745187, ...
%p b:= proc(l, t) option remember; local n, s;
%p n, s:= nops(l), add(i, i=l);
%p `if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>
%p `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p end:
%p A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, l}]};
%t 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 *)
%Y 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.
%K nonn,tabl
%O 0,34
%A _Alois P. Heinz_, Jul 02 2012