A238020 Number of nonconsecutive chess tableaux with n cells.

1, 1, 1, 1, 2, 2, 4, 5, 10, 15, 33, 52, 126, 213, 537, 991, 2563, 5118, 13670, 29171, 81069, 180813, 525755, 1216996, 3693934, 8843831, 27797975, 69106326, 223116931, 577433770, 1903516721, 5136516772, 17257698892, 48388514996, 166022450140, 481137194184

Offset: 0

Author

Alois P. Heinz, Feb 17 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(6) = 4:
[1]   [1 6]   [1 4]   [1 4]
[2]   [2]     [2 5]   [2 5]
[3]   [3]     [3]     [3 6]
[4]   [4]     [6]
[5]   [5]
[6]
		

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. A214088, A214459, A214460, A214461, A237770, A238014, A238184.

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$n], 0), `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$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 = 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 -> If[i == n && l[[n]] == 1, Nothing, l[[i]] - 1]], i], 0], {i, 1, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, b[Join[l, Table[1, n]], 0], 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, {}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Nov 08 2017, after Alois P. Heinz *)