A214087 Sum of the squares of numbers of nonconsecutive tableaux over all partitions of n.

1, 1, 1, 2, 6, 21, 92, 489, 3000, 20970, 166714, 1467337, 14212491, 149992662, 1723338952, 21393028409, 285061374438, 4054622024814, 61301381208116, 982904573560309, 16672187358390360, 298389960090957330, 5617735345244596804, 110942937545014894799

Offset: 0

Author

Alois P. Heinz, Jul 02 2012

Keywords

• nonn

Comments

A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..40
• T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
• Wikipedia, Young tableau

Crossrefs

Cf. A108774, A237770.

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 l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=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-> `if`(n<2, 1, g(n, n, [])):
seq(a(n), n=0..20);

Mathematica

b[l_, t_] := b[l, t] = Module[{n = Length[l], s = Total[l]}, If[s == 0, 1, Sum[If[t != i && l[[i]] > If[i == n, 0, l[[i + 1]]], b[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, b[Join[l, Table[1, n]], 0]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Table[i, j]]], {j, 0, n/i}]]];
a[n_] := If[n < 2, 1, g[n, n, {}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 23 2018, translated from Maple *)