A214021 -id:A214021 - cp's OEIS frontend

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, 5583, 19683, 72162, 274796, 1082439, 4406706, 18484332, 79818616, 353995743, 1611041726, 7510754022, 35842380314, 174850257639, 871343536591, 4430997592209, 22978251206350, 121410382810005, 653225968918521

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 13 2014

nonn

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
Also the number of ballot sequences without two consecutive elements equal. A ballot sequence B is a string such that, for all prefixes P of B, h(i)>=h(j) for iA000085).
First column (k=0) of A238125.

			The a(5) = 9 such tableaux of 5 are:
[1]   [2]  [3]   [4]  [5]  [6]  [7]  [8]  [9]
135   13   135   13   13   14   14   15   1
24    24   2     25   2    25   2    2    2
      5    4     4    4    3    3    3    3
                      5         5    4    4
                                          5
The corresponding ballot sequences are:
1:  [ 0 1 0 1 0 ]
2:  [ 0 1 0 1 2 ]
3:  [ 0 1 0 2 0 ]
4:  [ 0 1 0 2 1 ]
5:  [ 0 1 0 2 3 ]
6:  [ 0 1 2 0 1 ]
7:  [ 0 1 2 0 3 ]
8:  [ 0 1 2 3 0 ]
9:  [ 0 1 2 3 4 ]

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..68 (terms 0..48 from Alois P. Heinz)
Timothy Y. Chow, Henrik Eriksson and C. Kenneth Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol.11, no.2, (2005).
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Cf. A000085 (all Young tableaux), A000957, A001181, A214021, A214087, A214159, A214875.
Cf. A238126 (tableaux with one succession), A238127 (two successions).
Cf. A264051, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [i, l[]]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..30);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember;
      `if`(n<1, 1, add(`if`(i<>v and (i=1 or l[i-1]>l[i]),
       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
       b(n-1, nops(l)+1, [l[], 1]))
    end:
a:= proc(n) option remember; forget(b); b(n-1, 1, [1]) end:
seq(a(n), n=0..30);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i != v && (i == 1 || l[[i-1]] > l[[i]]), b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; a[n_] := a[n] = b[n-1, 1, {1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, translated from 2nd Maple program *)

a(n) = Sum_{k=1..A264078(n)} k * A264051(n,k). - Alois P. Heinz, Nov 02 2015

A214088 Number A(n,k) of n X k nonconsecutive 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, 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 *)

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 *)

A214159 Number of n X 3 nonconsecutive tableaux.

Original entry on oeis.org

1, 0, 1, 6, 72, 960, 14257, 228738, 3896062, 69590192, 1291977768, 24767357824, 487787117424, 9830929279248, 202119964701201, 4228355224663002, 89820746194643154, 1934075860467190608, 42153272431384984926, 928785495190695948612, 20666589509182675178064

Offset: 0

Alois P. Heinz, Jul 05 2012

nonn

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

			a(2) = 1:
  [1 3 5]
  [2 4 6].
a(3) = 6:
  [1 4 7]   [1 3 7]   [1 4 6]   [1 3 6]   [1 3 6]   [1 3 5]
  [2 5 8]   [2 5 8]   [2 5 8]   [2 5 8]   [2 4 8]   [2 6 8]
  [3 6 9]   [4 6 9]   [3 7 9]   [4 7 9]   [5 7 9]   [4 7 9].

Alois P. Heinz, Table of n, a(n) for n = 0..700
T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
Wikipedia, Young tableau

Column k=3 of 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 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..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 0, 1][n+1],
      ((22120*n^5 -58226*n^4 +31547*n^3 +42158*n^2 -53589*n +16146)*a(n-1)
      -(5988 -27068*n +50360*n^2 -62154*n^3 +49630*n^4 -15400*n^5)*a(n-2)
      -(7560*n^5 -23058*n^4 +24333*n^3 -11976*n^2 +2967*n -306)*a(n-3))
      /(6*(n+2)*(140*n^2-357*n+234)*(n+1)^2))
    end:
seq(a(n), n=0..25);

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}]]];
a[n_] := b[Table[3, n], 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from first Maple program *)

a(n) ~ 3^(3*n+13/2) / (2^12 * Pi * n^4). - Vaclav Kotesovec, Jul 16 2014

A214875 Number of 4 X n nonconsecutive tableaux.

Original entry on oeis.org

1, 1, 6, 72, 1289, 29889, 831174, 26455564, 934766625, 35896627737, 1475461220832, 64175536953702, 2928055871469177, 139180794974903769, 6854741942660442918, 348244986517582367748, 18183302860592129336633, 972820066413029570529513, 53192593416458179801289034

Offset: 0

Alois P. Heinz, Jul 28 2012

nonn

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

			a(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].

Alois P. Heinz, Table of n, a(n) for n = 0..534
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

Row n=4 of A214021.

a:= proc(n) option remember; `if`(n<3, [1,1,6][n+1], ((1620*n^7
       -13770*n^6 +41958*n^5 -48762*n^4 -6642*n^3 +62532*n^2 -48600*n
       +11664)*a(n-3) +(-3260*n^7 +11360*n^6 -4169*n^5 -19015*n^4
       +14521*n^3 +6179*n^2 -7380*n +1764)*a(n-2) +(-80*n +1964*n^3
       -7469*n^4 +3631*n^2 -5236*n^5 +2590*n^6 +1660*n^7 -300)*a(n-1))
       /((2*n+3)*(n+3)*(10*n^2-15*n-1)*(n+2)^3))
    end:
seq(a(n), n=0..25);

a[0] = a[1] = 1; a[2] = 6; a[n_] := a[n] = ((1620n^7 - 13770n^6 + 41958n^5 - 48762n^4 - 6642n^3 + 62532n^2 - 48600n + 11664) a[n-3] + (-3260n^7 + 11360n^6 - 4169n^5 - 19015n^4 + 14521n^3 + 6179n^2 - 7380n + 1764) a[n-2] + (-80n + 1964n^3 - 7469n^4 + 3631n^2 - 5236n^5 + 2590n^6 + 1660n^7 - 300) a[n-1])/((2n + 3)(n + 3)(10n^2 - 15n - 1)(n + 2)^3);
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

a(n) ~ 3^(4*n+12) / (2^13 * Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Jul 16 2014

A264103 Number of n X n nonconsecutive tableaux.

Original entry on oeis.org

1, 1, 1, 6, 1289, 13652068, 11865331748843, 1232033659827201777222, 20955050449849509663209289613921, 76615072242390448445916336191834325715261848, 76456972050113830615729276134092575545874371011199394401950, 25770844284993968943713846068617488831241440984966512955013952234546614462044

Offset: 0

Alois P. Heinz, Nov 03 2015

nonn

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

			a(3) = 6:
  [1 4 7]  [1 3 7]  [1 4 6]  [1 3 6]  [1 3 6]  [1 3 5]
  [2 5 8]  [2 5 8]  [2 5 8]  [2 5 8]  [2 4 8]  [2 6 8]
  [3 6 9]  [4 6 9]  [3 7 9]  [4 7 9]  [5 7 9]  [4 7 9].

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

Main diagonal of A214021.

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}]]];
a[n_] := a[n] = If[n<1, 1, b[Array[n&, n], 0]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 11}] (* Jean-François Alcover, Sep 08 2021, after Alois P. Heinz in A214021 *)

a(n) = A214021(n,n).

cp's OEIS Frontend

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

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

A214159 Number of n X 3 nonconsecutive tableaux.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A214875 Number of 4 X n nonconsecutive tableaux.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264103 Number of n X n nonconsecutive tableaux.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula