A214460 Number of 4 X 2*n+1 nonconsecutive chess tableaux.

1, 1, 1, 1, 2, 9, 44, 233, 1472, 10610, 82828, 688511, 6042580, 55523953, 530498426, 5242548633, 53361716373, 557495995958, 5961086843092, 65077096683358, 723862991146939, 8189257485453446, 94087399914391254, 1096331953010025684, 12941148147430923798

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..70 from Alois P. Heinz)

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) ~ c * 16^n / n^(15/2), where c = 5.347555... - Vaclav Kotesovec, Dec 05 2017

Example

a(4) = 2:
  [1, 4,  7, 12, 15, 20, 23, 28, 31]  [1,   4,  7, 10, 13, 16, 19, 22, 25]
  [2, 5, 10, 13, 18, 21, 26, 29, 34]  [2,   5,  8, 11, 14, 17, 28, 31, 34]
  [3, 8, 11, 16, 19, 24, 27, 32, 35]  [3,   6,  9, 20, 23, 26, 29, 32, 35]
  [6, 9, 14, 17, 22, 25, 30, 33, 36]  [12, 15, 18, 21, 24, 27, 30, 33, 36].

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=l[i]-1, l), i), 0), i=1..n))
    end:
a:= n-> b([(2*n+1)$4], 0):
seq(a(n), n=0..25);

Mathematica

b[l_List, 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 -> l[[i]]-1], i], 0], {i, 1, n}]]]; a[n_] := b[{2n+1, 2n+1, 2n+1, 2n+1}, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, translated from Maple *)

Crossrefs

Bisection (odd part) of row n=4 of A214088.

Sequence in context: A364476 A339440 A026302 * A124889 A317134 A295809

Adjacent sequences: A214457 A214458 A214459 * A214461 A214462 A214463

Keyword

nonn

Author

Alois P. Heinz, Jul 18 2012

Status

approved