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A238184 Sum of the squares of numbers of nonconsecutive chess tableaux over all partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

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

FORMULA

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

EXAMPLE

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]    :              :         :         :

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 *)

CROSSREFS

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

Sequence in context: A052949 A014266 A032441 * A065844 A131199 A112059

Adjacent sequences:  A238181 A238182 A238183 * A238185 A238186 A238187

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 19 2014

STATUS

approved

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Last modified February 20 14:14 EST 2020. Contains 332078 sequences. (Running on oeis4.)