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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.

%I A214088 #31 Oct 05 2018 20:10:02
%S A214088 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 A214088 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 A214088 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 A214088 Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A214088 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 A214088 Alois P. Heinz, <a href="/A214088/b214088.txt">Antidiagonals n = 0..21, flattened</a>
%H A214088 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 A214088 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 A214088 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A214088 A(3,5) = 1:
%e A214088   [1 4 7 10 13]
%e A214088   [2 5 8 11 14]
%e A214088   [3 6 9 12 15].
%e A214088 A(7,2) = 5:
%e A214088   [1  8]   [1  6]   [1  4]   [1   6]   [1   4]
%e A214088   [2  9]   [2  7]   [2  5]   [2   7]   [2   5]
%e A214088   [3 10]   [3 10]   [3 10]   [3   8]   [3   8]
%e A214088   [4 11]   [4 11]   [6 11]   [4   9]   [6   9]
%e A214088   [5 12]   [5 12]   [7 12]   [5  12]   [7  12]
%e A214088   [6 13]   [8 13]   [8 13]   [10 13]   [10 13]
%e A214088   [7 14]   [9 14]   [9 14]   [11 14]   [11 14].
%e A214088 Square array A(n,k) begins:
%e A214088   1,  1,  1,   1,    1,      1,        1,          1, ...
%e A214088   1,  1,  0,   0,    0,      0,        0,          0, ...
%e A214088   1,  1,  0,   0,    0,      0,        0,          0, ...
%e A214088   1,  1,  1,   1,    1,      1,        1,          1, ...
%e A214088   1,  1,  0,   1,    0,      1,        0,          1, ...
%e A214088   1,  1,  2,   7,   35,    212,     1421,      10128, ...
%e A214088   1,  1,  0,  27,    0,   5075,        0,    2402696, ...
%e A214088   1,  1,  5, 128, 6212, 430275, 42563460, 5601745187, ...
%p A214088 b:= proc(l, t) option remember; local n, s;
%p A214088        n, s:= nops(l), add(i, i=l);
%p A214088       `if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>
%p A214088       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214088     end:
%p A214088 A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
%p A214088 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A214088 b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, l}]};
%t A214088 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 A214088 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 A214088 nonn,tabl
%O A214088 0,34
%A A214088 _Alois P. Heinz_, Jul 02 2012