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A238014 Number of chess tableaux with n cells. 2
1, 1, 2, 2, 4, 6, 12, 20, 48, 84, 216, 408, 1104, 2280, 6288, 14128, 40256, 96240, 287904, 714016, 2246592, 5750112, 18900672, 49973568, 169592576, 466175808, 1618212224, 4637091200, 16393123072, 48926588544, 176264622336, 545058738944, 2008508679168 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A standard Young tableau (SYT) with cell(i,j) + i + j == 1 mod 2 for all cells is called a chess tableau. In other words, the odd numbered cells appear in the first, third, fifth, etc., skew diagonal, and the even numbered cells appear in the second, fourth, sixth, etc., skew diagonal. The definition appears first in the article by Jonas Sjöstrand.

All terms for n>=2 are even, as the conjugate of each chess tableau is a different chess tableau for n>=2.

Number of ballot sequences (with least element and first index either both 0 or both 1) with index of first occurrence of each element e of same parity as e, and identical elements separated by an even number of different elements, see example. [Joerg Arndt, Feb 28 2014]

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..58

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)} chess(lambda), where chess(lambda) is the number of chess tableaux of shape lambda.

EXAMPLE

a(5) = 6:

[1]  [1 4]  [1 2 3]  [1 4 5]  [1 2 3]  [1 2 3 4 5]

[2]  [2 5]  [4]      [2]      [4 5]

[3]  [3]    [5]      [3]

[4]

[5]

Note how the tableaux become partial chessboards when reduced modulo 2:

[1]  [1 0]  [1 0 1]  [1 0 1]  [1 0 1]  [1 0 1 0 1]

[0]  [0 1]  [0]      [0]      [0 1]

[1]  [1]    [1]      [1]

[0]

[1]

From Joerg Arndt, Feb 28 2014: (Start)

The a(7) = 20 ballot sequences are (dots for zeros):

01:    [ . . . . . . . ]

02:    [ . . . . . 1 1 ]

03:    [ . . . . . 1 2 ]

04:    [ . . . 1 1 . . ]

05:    [ . . . 1 1 . 2 ]

06:    [ . . . 1 1 1 2 ]

07:    [ . . . 1 2 . . ]

08:    [ . . . 1 2 . 1 ]

09:    [ . . . 1 2 3 1 ]

10:    [ . . . 1 2 3 4 ]

11:    [ . 1 2 . . . . ]

12:    [ . 1 2 . . . 1 ]

13:    [ . 1 2 . . 3 1 ]

14:    [ . 1 2 . . 3 4 ]

15:    [ . 1 2 . 1 2 . ]

16:    [ . 1 2 . 1 3 . ]

17:    [ . 1 2 . 1 3 4 ]

18:    [ . 1 2 3 4 . . ]

19:    [ . 1 2 3 4 . 1 ]

20:    [ . 1 2 3 4 5 6 ]

(End)

MAPLE

b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,

      `if`(args[nargs]=0, b(subsop(nargs=NULL, [args])[]),

      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), `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[args_] := b[args] = Module[{nargs = Length[args], s = Total[args]}, If[s == 0, 1, If[Last[args] == 0, b[Most[args]], Sum[If[Mod[s + i - args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i + 1]]], b[Append[ ReplacePart[ args, i -> args[[i]] - 1], 0]], 0], {i, 1, nargs}]]]];

g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, b[Join[l, Table[1, n]]], If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Table[i, j]]], {j, 0, n/i}]]];

a[n_] := g[n, n, {}];

Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Nov 14 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A108774, A214020, A237770, A238020.

Sequence in context: A003000 A216957 A122536 * A052953 A128209 A274935

Adjacent sequences:  A238011 A238012 A238013 * A238015 A238016 A238017

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 17 2014

STATUS

approved

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Last modified October 14 09:57 EDT 2019. Contains 327995 sequences. (Running on oeis4.)