A214020 - OEIS

%I #31 Oct 05 2018 20:06:53

%S 1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,2,0,1,1,1,1,2,6,6,2,

%T 1,1,1,1,0,22,0,22,0,1,1,1,1,5,92,324,324,92,5,1,1,1,1,0,422,0,8716,0,

%U 422,0,1,1,1,1,14,2074,47570,343234,343234,47570,2074,14,1,1

%N Number A(n,k) of n X k 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 is called a chess tableau.  The definition appears first in the article by Jonas Sjöstrand.

%H Alois P. Heinz, <a href="/A214020/b214020.txt">Antidiagonals n = 0..24, 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(4,3) = A(3,4) = 6:

%e   [1 4  7]  [1 4  5]  [1 2  3]  [1 4  7]  [ 1  4  7]  [ 1  2  3]

%e   [2 5 10]  [2 7 10]  [4 7 10]  [2 5 10]  [ 2  5  8]  [ 4  5  6]

%e   [3 8 11]  [3 8 11]  [5 8 11]  [3 6 11]  [ 3  6  9]  [ 7  8  9]

%e   [6 9 12]  [6 9 12]  [6 9 12]  [8 9 12]  [10 11 12]  [10 11 12].

%e Square array A(n,k) begins:

%e   1,  1,  1,   1,     1,        1,          1,             1, ...

%e   1,  1,  1,   1,     1,        1,          1,             1, ...

%e   1,  1,  0,   1,     0,        2,          0,             5, ...

%e   1,  1,  1,   2,     6,       22,         92,           422, ...

%e   1,  1,  0,   6,     0,      324,          0,         47570, ...

%e   1,  1,  2,  22,   324,     8716,     343234,      17423496, ...

%e   1,  1,  0,  92,     0,   343234,          0,    8364334408, ...

%e   1,  1,  5, 422, 47570, 17423496, 8364334408, 6873642982160, ...

%p b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,

%p       add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0,

%p       args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs))

%p     end:

%p A:= (n, k)-> `if`(n<k, A(k, n), `if`(k<2, 1, b(n$k))):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%t 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_ *)

%Y Cf. A000108 (bisection of row 2), A001181 (row 3), A108774, A214021, A214088.

%K nonn,tabl

%O 0,25

%A _Alois P. Heinz_, Jul 01 2012