OFFSET
0,5
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..200 (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
FORMULA
a(n) ~ c * 16^n / n^(15/2), where c = 5.347555... - Vaclav Kotesovec, Dec 05 2017
EXAMPLE
a(4) = 2:
[1, 4, 7, 12, 15, 20, 23, 28, 31] [1, 4, 7, 10, 13, 16, 19, 22, 25]
[2, 5, 10, 13, 18, 21, 26, 29, 34] [2, 5, 8, 11, 14, 17, 28, 31, 34]
[3, 8, 11, 16, 19, 24, 27, 32, 35] [3, 6, 9, 20, 23, 26, 29, 32, 35]
[6, 9, 14, 17, 22, 25, 30, 33, 36] [12, 15, 18, 21, 24, 27, 30, 33, 36].
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([(2*n+1)$4], 0):
seq(a(n), n=0..25);
MATHEMATICA
b[l_List, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], 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[{2n+1, 2n+1, 2n+1, 2n+1}, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 18 2012
STATUS
approved