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A214461
Number of 5 X n nonconsecutive chess tableaux.
4
1, 1, 2, 7, 35, 212, 1421, 10128, 75724, 593905, 4927764, 43884083, 426665219, 4591883867, 54943702996, 725753304587, 10428313247758, 160361700765626, 2602492613187389, 44111609147837212, 775162479337156853, 14049351313802046511, 261640973700411314373
OFFSET
0,3
COMMENTS
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.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..114 (terms 0..70 from Alois P. Heinz)
T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
Jonas Sjöstrand, On the sign-imbalance of partition shapes, arXiv:math/0309231v3 [math.CO], 2005.
Wikipedia, Young tableau
EXAMPLE
a(3) = 7:
[1 6 11] [1 4 11] [1 6 9] [1 4 9] [1 4 7] [1 4 7] [1 4 7]
[2 7 12] [2 5 12] [2 7 10] [2 5 10] [2 5 10] [2 5 10] [2 5 8]
[3 8 13] [3 8 13] [3 8 13] [3 8 13] [3 8 13] [3 6 13] [3 10 13]
[4 9 14] [6 9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
[5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].
MAPLE
b:= proc(l, t) option remember; local n, s;
n, s:= nops(l), add(i, i=l);
`if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>
`if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
end:
a:= n-> b([n$5], 0):
seq(a(n), n=0..25);
MATHEMATICA
b[l_List, t_] := b[l, t] = With[{n = Length[l], s = Total[l]}, 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_] := b[Table[n, {5}], 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, translated from Maple *)
CROSSREFS
Row n=5 of A214088.
Sequence in context: A024719 A086637 A172511 * A130458 A003575 A043546
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 18 2012
STATUS
approved