A214461 -id:A214461 - 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 *)

A238020 Number of nonconsecutive chess tableaux with n cells.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 10, 15, 33, 52, 126, 213, 537, 991, 2563, 5118, 13670, 29171, 81069, 180813, 525755, 1216996, 3693934, 8843831, 27797975, 69106326, 223116931, 577433770, 1903516721, 5136516772, 17257698892, 48388514996, 166022450140, 481137194184

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 where entries m and m+1 never appear in the same row is called a nonconsecutive chess tableau.

			a(6) = 4:
[1]   [1 6]   [1 4]   [1 4]
[2]   [2]     [2 5]   [2 5]
[3]   [3]     [3]     [3 6]
[4]   [4]     [6]
[5]   [5]
[6]

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. A214088, A214459, A214460, A214461, A237770, A238014, A238184.

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$n], 0), `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> g(n, n, []):
seq(a(n), n=0..32);

b[l_, t_] := b[l, t] = Module[{ n = Length[l], s = 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_] := If[n == 0 || i == 1, b[Join[l, Table[1, n]], 0], 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 08 2017, after Alois P. Heinz *)

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.

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

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A238020 Number of nonconsecutive 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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