A108774 -id:A108774 - cp's OEIS frontend

A214020 Number A(n,k) of n X k chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 0, 22, 0, 22, 0, 1, 1, 1, 1, 5, 92, 324, 324, 92, 5, 1, 1, 1, 1, 0, 422, 0, 8716, 0, 422, 0, 1, 1, 1, 1, 14, 2074, 47570, 343234, 343234, 47570, 2074, 14, 1, 1

Offset: 0

Alois P. Heinz, Jul 01 2012

A standard Young tableau (SYT) with cell(i,j)+i+j == 1 mod 2 for all cells is called a chess tableau. The definition appears first in the article by Jonas Sjöstrand.

			A(4,3) = A(3,4) = 6:
  [1 4  7]  [1 4  5]  [1 2  3]  [1 4  7]  [ 1  4  7]  [ 1  2  3]
  [2 5 10]  [2 7 10]  [4 7 10]  [2 5 10]  [ 2  5  8]  [ 4  5  6]
  [3 8 11]  [3 8 11]  [5 8 11]  [3 6 11]  [ 3  6  9]  [ 7  8  9]
  [6 9 12]  [6 9 12]  [6 9 12]  [8 9 12]  [10 11 12]  [10 11 12].
Square array A(n,k) begins:
  1,  1,  1,   1,     1,        1,          1,             1, ...
  1,  1,  1,   1,     1,        1,          1,             1, ...
  1,  1,  0,   1,     0,        2,          0,             5, ...
  1,  1,  1,   2,     6,       22,         92,           422, ...
  1,  1,  0,   6,     0,      324,          0,         47570, ...
  1,  1,  2,  22,   324,     8716,     343234,      17423496, ...
  1,  1,  0,  92,     0,   343234,          0,    8364334408, ...
  1,  1,  5, 422, 47570, 17423496, 8364334408, 6873642982160, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened
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

Cf. A000108 (bisection of row 2), A001181 (row 3), A108774, A214021, A214088.

b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1,
      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:
A:= (n, k)-> `if`(n

b[args_List] := b[args] = Module[{s = Total[args], nargs = Length[args]}, If[s == 0, 1, Sum[If[Mod[s + i - args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i + 1]]], b[ReplacePart[args, i -> args[[i]] - 1]], 0], {i, 1, nargs}]]]; A[n_, k_] := If[n < k, A[k, n], If[k < 2, 1, b[Array[n &, k]]]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)

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

Original entry on oeis.org

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

Alois P. Heinz, Jul 02 2012

nonn

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

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

Cf. A108774, A237770.

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);

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 *)

A238014 Number of chess tableaux with n cells.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Feb 17 2014

nonn

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]

			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)

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

Cf. A108774, A214020, A237770, A238020.

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);

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 *)

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

A238184 Sum of the squares of numbers of nonconsecutive chess tableaux over all partitions of n.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Feb 19 2014

nonn

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.

			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]    :              :         :         :

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

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

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);

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 *)

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

cp's OEIS Frontend

A214020 Number A(n,k) of n X k chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A238014 Number of chess tableaux with n cells.

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A238184 Sum of the squares of numbers of nonconsecutive chess tableaux over all partitions of n.

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