A237770 -id:A237770 - cp's OEIS frontend

A238125 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k flat steps, n>=0, 0<=k<=n.

1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 9, 8, 6, 2, 1, 0, 22, 24, 17, 9, 3, 1, 0, 59, 70, 57, 29, 13, 3, 1, 0, 170, 224, 191, 108, 49, 17, 4, 1, 0, 516, 744, 663, 399, 201, 69, 23, 4, 1, 0, 1658, 2588, 2415, 1573, 802, 322, 104, 28, 5, 1, 0, 5583, 9317, 9108, 6249, 3343, 1408, 510, 137, 35, 5, 1, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Also number of standard Young tableaux with n cells and exactly k successions. A succession is a pair of cells (v, v+1) lying in the same row.
Columns k=0-10 give: A237770, A238126, A238127, A241774, A241775, A241776, A241777, A241778, A241779, A241780, A241781.
T(2n,n) gives A241785.
Row sums are A000085.

			Triangle starts:
00:     1;
01:     1,     0;
02:     1,     1,     0;
03:     2,     1,     1,     0;
04:     4,     3,     2,     1,     0;
05:     9,     8,     6,     2,     1,    0;
06:    22,    24,    17,     9,     3,    1,    0;
07:    59,    70,    57,    29,    13,    3,    1,   0;
08:   170,   224,   191,   108,    49,   17,    4,   1,   0;
09:   516,   744,   663,   399,   201,   69,   23,   4,   1,  0;
10:  1658,  2588,  2415,  1573,   802,  322,  104,  28,   5,  1, 0;
11:  5583,  9317,  9108,  6249,  3343, 1408,  510, 137,  35,  5, 1, 0;
12: 19683, 34924, 35695, 25642, 14368, 6440, 2411, 751, 189, 42, 6, 1, 0;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..45, flattened

b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
      add(`if`(i=1 or l[i-1]>l[i], `if`(i=v, x, 1)*
      b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
      b(n-1, nops(l)+1, [l[], 1])))
    end:
T:= n-> seq(coeff(b(n-1, 1, [1]), x, i), i=0..n):
seq(T(n), n=0..12);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i == v, x, 1]*b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; T[n_] := Table[Coefficient[b[n-1, 1, {1}], x, i], {i, 0, n}]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)

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 *)

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 *)

A238126 Number of standard Young tableaux with n cells and exactly one succession.

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 24, 70, 224, 744, 2588, 9317, 34924, 135297, 542123, 2236834, 9508297, 41511215, 186109781, 854874944, 4021672983, 19344343843, 95093249014, 477137036748, 2442413708120, 12742038926613, 67714763161526, 366266085720565, 2015454903261855

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.
Also number of ballot sequences having one flat step.
Second column (k=1) of A238125.

Cf. A237770 (no successions), A238127 (two successions).

A238127 Number of standard Young tableaux with n cells and exactly two successions.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 17, 57, 191, 663, 2415, 9108, 35695, 143989, 599802, 2566917, 11298164, 50967216, 235745644, 1115324000, 5397332497, 26669487517, 134528555379, 691856601631, 3626390958551, 19353306241764, 105122093620388, 580689432523534, 3260906342453966

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

nonn

A succession is a pair of cells (v, v+1) lying in the same row.
Also number of ballot sequences having two flat steps.
Third column (k=2) of A238125.

Cf. A237770 (no successions), A238126 (one succession).

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 0, 2, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 7, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 7, 3, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

Column k=0 gives A025065(n-2) for n>=2.

			Triangle begins:
0,1,
0,1,
1,1,
1,2,
2,2,1,
2,3,0,2,
4,2,1,1,1,1,1,
4,3,1,0,0,2,2,0,1,0,1,0,0,0,1,
7,2,0,0,1,0,3,0,1,0,2,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
...

Alois P. Heinz, Rows n = 0..14, flattened
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
FindStat - Combinatorial Statistic Finder, Number of standard Young tableaux of an integer partition such that no k and k+1 appear in the same row.

Cf. A000041, A001181, A237770, 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, x^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:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 02 2015

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := If[n == 0 || i == 1, x^h[Join[Array[1 &, n], l], 0], If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Join[{i}, l]]] ]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Sum_{k=1..A264078(n)} k*T(n,k) = A237770(n). - Alois P. Heinz, Nov 02 2015

A264078 The maximal number of standard Young tableaux without a succession v, v+1 in a row that a single partition of n can have.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 14, 30, 76, 170, 553, 1583, 5106, 14090, 41002, 164769, 603513, 2418348, 8335804, 28704417, 109618261, 466318442, 2114095511, 10276979159, 43213859606, 175668903294, 793946150358, 3490939879402, 15500974371599, 82490059523125

Offset: 0

Alois P. Heinz, Nov 02 2015

nonn

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.

			a(6) = 6: partition [2,2,1,1] has 6 standard Young tableaux without a succession v, v+1 in a row, which is maximal for a partition of n=6:
15   14   14   13   13   13
26   26   25   26   25   24
3    3    3    4    4    5
4    5    6    5    6    6

Alois P. Heinz, Table of n, a(n) for n = 0..60
Wikipedia, Young tableau

Cf. A237770, A264051.

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, max(g(n, i-1, l),
      `if`(i>n, 0, g(n-i, i, [i, l[]])))))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2015

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, h[Join[Array[1 &, n], l], 0], If[i < 1, 0, Max[g[n, i - 1, l], If[i > n, 0, g[n - i, i, Join[{i}, l]]]]]]; a[n_] := g[n, n, {}];  Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

a(n) = max { k : A264051(n,k) > 0 }.

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

A238125 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k flat steps, n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A238020 Number of nonconsecutive chess tableaux with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A238126 Number of standard Young tableaux with n cells and exactly one succession.

Views

Author

Keywords

Comments

Crossrefs

A238127 Number of standard Young tableaux with n cells and exactly two successions.

Views

Author

Keywords

Comments

Crossrefs

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264078 The maximal number of standard Young tableaux without a succession v, v+1 in a row that a single partition of n can have.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A238014 Number of chess tableaux with n cells.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula