A214020 -id:A214020 - cp's OEIS frontend

A214088 Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 7, 0, 1, 1, 1, 0, 0, 1, 1, 35, 27, 5, 1, 1, 1, 0, 0, 1, 0, 212, 0, 128, 0, 1, 1, 1, 0, 0, 1, 1, 1421, 5075, 6212, 640, 14, 1, 1, 1, 0, 0, 1, 0, 10128, 0, 430275, 0, 3351, 0, 1, 1

Offset: 0

Alois P. Heinz, Jul 02 2012

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(3,5) = 1:
  [1 4 7 10 13]
  [2 5 8 11 14]
  [3 6 9 12 15].
A(7,2) = 5:
  [1  8]   [1  6]   [1  4]   [1   6]   [1   4]
  [2  9]   [2  7]   [2  5]   [2   7]   [2   5]
  [3 10]   [3 10]   [3 10]   [3   8]   [3   8]
  [4 11]   [4 11]   [6 11]   [4   9]   [6   9]
  [5 12]   [5 12]   [7 12]   [5  12]   [7  12]
  [6 13]   [8 13]   [8 13]   [10 13]   [10 13]
  [7 14]   [9 14]   [9 14]   [11 14]   [11 14].
Square array A(n,k) begins:
  1,  1,  1,   1,    1,      1,        1,          1, ...
  1,  1,  0,   0,    0,      0,        0,          0, ...
  1,  1,  0,   0,    0,      0,        0,          0, ...
  1,  1,  1,   1,    1,      1,        1,          1, ...
  1,  1,  0,   1,    0,      1,        0,          1, ...
  1,  1,  2,   7,   35,    212,     1421,      10128, ...
  1,  1,  0,  27,    0,   5075,        0,    2402696, ...
  1,  1,  5, 128, 6212, 430275, 42563460, 5601745187, ...

Alois P. Heinz, Antidiagonals n = 0..21, 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 column k=2 for n>0), A214459 (column k=3), A214460 (bisection of row n=4), A214461 (row n=5), A214020, A214021.

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, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, 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_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 6, 6, 1, 1, 1, 0, 1, 22, 72, 18, 1, 1, 1, 0, 1, 92, 1289, 960, 57, 1, 1, 1, 0, 1, 422, 29889, 93964, 14257, 186, 1, 1, 1, 0, 1, 2074, 831174, 13652068, 8203915, 228738, 622, 1, 1

Offset: 0

Alois P. Heinz, Jul 01 2012

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

			A(2,4) = 1:
  [1 3 5 7]
  [2 4 6 8].
A(4,2) = 6:
  [1, 5]   [1, 4]   [1, 3]   [1, 4]   [1, 3]   [1, 3]
  [2, 6]   [2, 6]   [2, 6]   [2, 5]   [2, 5]   [2, 4]
  [3, 7]   [3, 7]   [4, 7]   [3, 7]   [4, 7]   [5, 7]
  [4, 8]   [5, 8]   [5, 8]   [6, 8]   [6, 8]   [6, 8].
Square array A(n,k) begins:
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  0,     0,       0,          0,              0, ...
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  2,     6,      22,         92,            422, ...
  1, 1,  6,    72,    1289,      29889,         831174, ...
  1, 1, 18,   960,   93964,   13652068,     2621897048, ...
  1, 1, 57, 14257, 8203915, 8134044455, 11865331748843, ...

Alois P. Heinz, Antidiagonals n = 0..23, flattened
T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Rows n=0+2, 3-4 give: A000012, A001181(k) for k>0, A214875.
Columns k=0+1, 2, 3 give: A000012, A000957(n+1), A214159.
Main diagonal gives A264103.
Cf. A214020, A214088.

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:
A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, 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}]] ] ; a[n_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, 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.

cp's OEIS Frontend

A214088 Number A(n,k) of n X k nonconsecutive chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A238014 Number of chess tableaux with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula