A238184 Sum of the squares of numbers of nonconsecutive chess tableaux over all partitions of n.

1, 1, 1, 1, 2, 2, 4, 7, 16, 37, 107, 282, 1020, 2879, 12507, 39347, 179231, 687974, 3225246, 14955561, 75999551, 392585613, 2271201137, 12183159188, 81562521256, 446611878413, 3336304592155, 19202329389234, 152803821604669, 958953289839930, 7835058287650579

Offset: 0

Author

Alois P. Heinz, Feb 19 2014

Keywords

• nonn

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.

Examples

			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]    :              :         :         :
		

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

Crossrefs

Cf. A108774, A214088, A214459, A214460, A214461, A238020.

Programs

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 *)

Formula

a(n) = Sum_{lambda : partitions(n)} ncc(lambda)^2, where ncc(k) is the number of nonconsecutive chess tableaux of shape k.