A214088 -id:A214088 - cp's OEIS frontend

A214021 Number A(n,k) of n X k nonconsecutive 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, 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 *)

A214020 Number A(n,k) of n X k chess 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, 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 *)

A214459 Number of n X 3 nonconsecutive chess tableaux.

Original entry on oeis.org

1, 0, 0, 1, 1, 7, 27, 128, 640, 3351, 18313, 103404, 600538, 3571717, 21683185, 134005373, 841259885, 5355078350, 34512405410, 224908338137, 1480420941781, 9833512593113, 65860442383487, 444453988418791, 3020274890688447, 20656019108074552, 142107550142684602

Offset: 0

Alois P. Heinz, Jul 18 2012

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(5) = 7:
  [1  6 11] [1  4 11] [1  6  9] [1  4  9] [1  4  7] [1  4  7] [1  4  7]
  [2  7 12] [2  5 12] [2  7 10] [2  5 10] [2  5 10] [2  5 10] [2  5  8]
  [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  6 13] [3 10 13]
  [4  9 14] [6  9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
  [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].

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

Column k=3 of 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 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([3$n], 0):
seq(a(n), n=0..25);

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_] := If[n < 1, 1, b[Array[3&, n], 0]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 13 2017, after Alois P. Heinz *)

a(n) ~ c * 8^n / n^4, where c = 0.250879571... - Vaclav Kotesovec, Sep 06 2017

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

Original entry on oeis.org

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

Alois P. Heinz, Jul 18 2012

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(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].

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

Bisection (odd part) of row n=4 of 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 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);

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 *)

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

A214461 Number of 5 X n nonconsecutive chess tableaux.

Original entry on oeis.org

1, 1, 2, 7, 35, 212, 1421, 10128, 75724, 593905, 4927764, 43884083, 426665219, 4591883867, 54943702996, 725753304587, 10428313247758, 160361700765626, 2602492613187389, 44111609147837212, 775162479337156853, 14049351313802046511, 261640973700411314373

Offset: 0

Alois P. Heinz, Jul 18 2012

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(3) = 7:
  [1  6 11] [1  4 11] [1  6  9] [1  4  9] [1  4  7] [1  4  7] [1  4  7]
  [2  7 12] [2  5 12] [2  7 10] [2  5 10] [2  5 10] [2  5 10] [2  5  8]
  [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  6 13] [3 10 13]
  [4  9 14] [6  9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
  [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].

Vaclav Kotesovec, Table of n, a(n) for n = 0..114 (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

Row n=5 of 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 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([n$5], 0):
seq(a(n), n=0..25);

b[l_List, t_] := b[l, t] = With[{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 -> l[[i]] - 1], i], 0], {i, 1, n}]]]; a[n_] := b[Table[n, {5}], 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017, 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.

cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A214459 Number of n X 3 nonconsecutive chess tableaux.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A214461 Number of 5 X n nonconsecutive chess tableaux.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A238020 Number of nonconsecutive chess tableaux with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica