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A238014
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Number of chess tableaux with n cells.
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2
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1, 1, 2, 2, 4, 6, 12, 20, 48, 84, 216, 408, 1104, 2280, 6288, 14128, 40256, 96240, 287904, 714016, 2246592, 5750112, 18900672, 49973568, 169592576, 466175808, 1618212224, 4637091200, 16393123072, 48926588544, 176264622336, 545058738944, 2008508679168
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OFFSET
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0,3
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COMMENTS
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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.
All terms for n>=2 are even, as the conjugate of each chess tableau is a different chess tableau for n>=2.
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]
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LINKS
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T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
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FORMULA
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a(n) = Sum_{lambda : partitions(n)} chess(lambda), where chess(lambda) is the number of chess tableaux of shape lambda.
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EXAMPLE
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a(5) = 6:
[1] [1 4] [1 2 3] [1 4 5] [1 2 3] [1 2 3 4 5]
[2] [2 5] [4] [2] [4 5]
[3] [3] [5] [3]
[4]
[5]
Note how the tableaux become partial chessboards when reduced modulo 2:
[1] [1 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1 0 1]
[0] [0 1] [0] [0] [0 1]
[1] [1] [1] [1]
[0]
[1]
The a(7) = 20 ballot sequences are (dots for zeros):
01: [ . . . . . . . ]
02: [ . . . . . 1 1 ]
03: [ . . . . . 1 2 ]
04: [ . . . 1 1 . . ]
05: [ . . . 1 1 . 2 ]
06: [ . . . 1 1 1 2 ]
07: [ . . . 1 2 . . ]
08: [ . . . 1 2 . 1 ]
09: [ . . . 1 2 3 1 ]
10: [ . . . 1 2 3 4 ]
11: [ . 1 2 . . . . ]
12: [ . 1 2 . . . 1 ]
13: [ . 1 2 . . 3 1 ]
14: [ . 1 2 . . 3 4 ]
15: [ . 1 2 . 1 2 . ]
16: [ . 1 2 . 1 3 . ]
17: [ . 1 2 . 1 3 4 ]
18: [ . 1 2 3 4 . . ]
19: [ . 1 2 3 4 . 1 ]
20: [ . 1 2 3 4 5 6 ]
(End)
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MAPLE
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b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
`if`(args[nargs]=0, b(subsop(nargs=NULL, [args])[]),
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:
g:= (n, i, l)-> `if`(n=0 or i=1, b(l[], 1$n), `if`(i<1, 0,
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> g(n, n, []):
seq(a(n), n=0..32);
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MATHEMATICA
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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}]]]];
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}]]];
a[n_] := g[n, n, {}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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