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

0,4

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

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 *)

Crossrefs

Cf. A108774, A237770.

Sequence in context: A115089 A304196 A266328 * A183950 A001928 A005638

Adjacent sequences: A214084 A214085 A214086 * A214088 A214089 A214090

Keyword

nonn

Author

Alois P. Heinz, Jul 02 2012

Status

approved