A108774 Sum of the squares of numbers of chess tableaux over all partitions of n.

1, 1, 2, 2, 4, 8, 16, 48, 160, 448, 2048, 6400, 31232, 125952, 604160, 3119104, 15638528, 93478912, 550141952, 3367698432, 24049516544, 146207539200, 1203934593024, 7615928598528, 67190404415488, 468355947429888, 4196459066949632, 33260378783744000

Offset: 0

Author

N. J. A. Sloane, Jun 27 2005

Keywords

• nonn

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.

Crossrefs

Cf. A214087.

Programs

Maple

b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
      add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0,
      args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, b(l[], 1$n)^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> `if`(n<2, 1, g(n, n, [])):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 01 2012

Mathematica

b[args_List] := b[args] = Module[{s=Total[args], nargs=Length[args]}, If[s == 0, 1, Sum[If[Mod[s+i-args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i+1]] ], b[ReplacePart[args, i -> args[[i]]-1]], 0], {i, 1, nargs}] ] ]; g[n_, i_, l_List] := g[n, i, l] = If[n == 0 || i == 1, b[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}] ] ]; a[n_] := If[n<2, 1, g[n, n, {}]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 08 2015, after Alois P. Heinz *)

Formula

a(n) = Sum_{lambda : partitions(n)} chess(lambda)^2, where chess(k) is the number of standard Young tableaux (SYT) with cell(i,j)+i+j == 1 mod 2. - Alois P. Heinz, Jun 30 2012

Extensions

More terms from Alois P. Heinz, Jun 30 2012