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%I #41 Nov 14 2017 08:20:06
%S 1,1,2,2,4,6,12,20,48,84,216,408,1104,2280,6288,14128,40256,96240,
%T 287904,714016,2246592,5750112,18900672,49973568,169592576,466175808,
%U 1618212224,4637091200,16393123072,48926588544,176264622336,545058738944,2008508679168
%N Number of chess tableaux with n cells.
%C A standard Young tableau (SYT) with cell(i,j) + i + j == 1 mod 2 for all cells is called a chess tableau. In other words, the odd numbered cells appear in the first, third, fifth, etc., skew diagonal, and the even numbered cells appear in the second, fourth, sixth, etc., skew diagonal. The definition appears first in the article by Jonas Sjöstrand.
%C All terms for n>=2 are even, as the conjugate of each chess tableau is a different chess tableau for n>=2.
%C Number of ballot sequences (with least element and first index either both 0 or both 1) with index of first occurrence of each element e of same parity as e, and identical elements separated by an even number of different elements, see example. [_Joerg Arndt_, Feb 28 2014]
%H Alois P. Heinz, <a href="/A238014/b238014.txt">Table of n, a(n) for n = 0..58</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>
%F a(n) = Sum_{lambda : partitions(n)} chess(lambda), where chess(lambda) is the number of chess tableaux of shape lambda.
%e a(5) = 6:
%e [1] [1 4] [1 2 3] [1 4 5] [1 2 3] [1 2 3 4 5]
%e [2] [2 5] [4] [2] [4 5]
%e [3] [3] [5] [3]
%e [4]
%e [5]
%e Note how the tableaux become partial chessboards when reduced modulo 2:
%e [1] [1 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1 0 1]
%e [0] [0 1] [0] [0] [0 1]
%e [1] [1] [1] [1]
%e [0]
%e [1]
%e From _Joerg Arndt_, Feb 28 2014: (Start)
%e The a(7) = 20 ballot sequences are (dots for zeros):
%e 01: [ . . . . . . . ]
%e 02: [ . . . . . 1 1 ]
%e 03: [ . . . . . 1 2 ]
%e 04: [ . . . 1 1 . . ]
%e 05: [ . . . 1 1 . 2 ]
%e 06: [ . . . 1 1 1 2 ]
%e 07: [ . . . 1 2 . . ]
%e 08: [ . . . 1 2 . 1 ]
%e 09: [ . . . 1 2 3 1 ]
%e 10: [ . . . 1 2 3 4 ]
%e 11: [ . 1 2 . . . . ]
%e 12: [ . 1 2 . . . 1 ]
%e 13: [ . 1 2 . . 3 1 ]
%e 14: [ . 1 2 . . 3 4 ]
%e 15: [ . 1 2 . 1 2 . ]
%e 16: [ . 1 2 . 1 3 . ]
%e 17: [ . 1 2 . 1 3 4 ]
%e 18: [ . 1 2 3 4 . . ]
%e 19: [ . 1 2 3 4 . 1 ]
%e 20: [ . 1 2 3 4 5 6 ]
%e (End)
%p b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
%p `if`(args[nargs]=0, b(subsop(nargs=NULL, [args])[]),
%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 g:= (n, i, l)-> `if`(n=0 or i=1, b(l[], 1$n), `if`(i<1, 0,
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p a:= n-> g(n, n, []):
%p seq(a(n), n=0..32);
%t b[args_] := b[args] = Module[{nargs = Length[args], s = Total[args]}, If[s == 0, 1, If[Last[args] == 0, b[Most[args]], Sum[If[Mod[s + i - args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i + 1]]], b[Append[ ReplacePart[ args, i -> args[[i]] - 1], 0]], 0], {i, 1, nargs}]]]];
%t g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, b[Join[l, Table[1, n]]], If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Table[i, j]]], {j, 0, n/i}]]];
%t a[n_] := g[n, n, {}];
%t Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Nov 14 2017, after _Alois P. Heinz_ *)
%Y Cf. A108774, A214020, A237770, A238020.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 17 2014
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