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
Alois P. Heinz, Table of n, a(n) for n = 0..50
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) = Sum_{lambda : partitions(n)} ncc(lambda)^2, where ncc(k) is the number of nonconsecutive chess tableaux of shape k.
EXAMPLE
a(7) = 1 + 2^2 + 1 + 1 = 7:
.
: [1111111] : [22111] : [3211] : [322] : <- shapes
:-----------+--------------+---------+---------:
: [1] : [1 6] [1 4] : [1 4 7] : [1 4 7] :
: [2] : [2 7] [2 5] : [2 5] : [2 5] :
: [3] : [3] [3] : [3] : [3 6] :
: [4] : [4] [6] : [6] : :
: [5] : [5] [7] : : :
: [6] : : : :
: [7] : : : :
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=`if`(i=n and l[n]=1, [][], l[i]-1), l), i), 0), i=1..n))
end:
g:= (n, i, l)-> `if`(n=0 or i=1, b([l[], 1$n], 0)^2, `if`(i<1, 0,
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> g(n, n, []):
seq(a(n), n=0..32);
MATHEMATICA
b[l_, 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 -> If[i==n && l[[n]]==1, Nothing, l[[i]]-1]], i], 0], {i, 1, n}]]]; g[n_, i_, l_] := g[n, i, l] = If[n==0 || i==1, b[Join[l, Array[1&, n]], 0]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Feb 17 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 19 2014
STATUS
approved