A215086 -id:A215086 - cp's OEIS frontend

A214753 Number T(n,k) of solid standard Young tableaux of n cells and height = k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 4, 4, 1, 0, 10, 16, 6, 1, 0, 26, 66, 34, 8, 1, 0, 76, 296, 192, 58, 10, 1, 0, 232, 1334, 1134, 406, 88, 12, 1, 0, 764, 6322, 6716, 2918, 730, 124, 14, 1, 0, 2620, 30930, 40872, 20718, 6118, 1186, 166, 16, 1, 0, 9496, 158008, 255308, 149826, 50056, 11310, 1796, 214, 18, 1

Offset: 0

Alois P. Heinz, Aug 02 2012

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    4,    1;
  0,  10,   16,    6,   1;
  0,  26,   66,   34,   8,  1;
  0,  76,  296,  192,  58, 10,  1;
  0, 232, 1334, 1134, 406, 88, 12,  1;

Alois P. Heinz, Rows n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-10 give: A000007(n), A000085(n) for n>0, A273582, A273583, A273584, A273585, A273586, A273587, A273588, A273589, A273590.
Diagonal and lower diagonal give: A000012, A005843.
Row sums give: A207542.
T(2n,n) gives A273591.
Cf. A215086.

b:= proc(n, k, l) option remember; `if`(n=0, 1,
       b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
T:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n-1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i-1]]], b[n-1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i-1, j]]) && (j == 1 || L[[i, j]] < L[[i, j-1]]), b[n-1, k, ReplacePart[L, i -> ReplacePart[ L[[i]], j -> L[[i, j]]+1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *)

T(n,k) = A215086(n,k) - A215086(n,k-1) for k>0, T(n,0) = A215086(n,0) = A000007(n).

A207542 Number of solid standard Young tableaux with n cells.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 633, 3207, 17589, 102627, 636033, 4161141, 28680717, 207318273, 1567344549, 12345147705, 101013795753, 856212871761, 7501911705747, 67815650852235, 631574151445665, 6051983918989833, 59605200185016639, 602764245172225251, 6252962956009863363

Offset: 0

Matthew C. Russell, Feb 24 2012

A solid standard Young tableaux (SSYT) with n cells is a way of placing the integers from 1 to n in a 3D Young diagram of a plane partition with the property that the entries increase from left to right, back to front, and bottom to top.
It is also the number of almost topological sequences (ATS) for the set N^3 at depth n with (N=set of nonnegative integers). See Balakrishnan et al. for definition and a proof of the bijection between SSYT and ATS. - Suresh Govindarajan, Mar 02 2012
Also the number of paths from a plane partition to the empty partition by repeated trimming. - Wouter Meeussen, Sep 03 2025

Shalosh B. Ekhad, Doron Zeilberger, and Vaclav Kotesovec, Table of n, a(n) for n = 0..37 (terms 0..30 from Shalosh B. Ekhad and Doron Zeilberger)
S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, J. Phys. A45 (2012) 055001; arXiv:1105.6231 [cond-mat.stat-mech], 2011.
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux.
Suresh Govindarajan, Almost Topological Sequences

Rows sums of A214753.
Main diagonal of A215086.
Column k=0 of A215120. - Alois P. Heinz, May 12 2014

b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n - 1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i - 1]]], b[n - 1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i - 1, j]]) && (j == 1 || L[[i, j]] < L[[i, j - 1]]), b[n - 1, k, ReplacePart[L, i -> ReplacePart[L[[i]], j -> L[[i, j]] + 1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
a[n_] := a[n] = Sum[T[n, k], {k, 0, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz in A214753 *)

A215120 Number T(n,k) of solid standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 33, 33, 23, 7, 1, 135, 135, 109, 43, 9, 1, 633, 633, 557, 261, 69, 11, 1, 3207, 3207, 2975, 1641, 507, 101, 13, 1, 17589, 17589, 16825, 10503, 3787, 869, 139, 15, 1, 102627, 102627, 100007, 69077, 28205, 7487, 1369, 183, 17, 1

Offset: 0

Alois P. Heinz, Aug 03 2012

			Triangle T(n,k) begins:
     1;
     1,    1;
     3,    3,    1;
     9,    9,    5,    1;
    33,   33,   23,    7,   1;
   135,  135,  109,   43,   9,   1;
   633,  633,  557,  261,  69,  11,  1;
  3207, 3207, 2975, 1641, 507, 101, 13,  1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012.
Wikipedia, Young tableau

Column k=0 gives: A207542.
Diagonal and lower diagonal give: A000012, A005408.
T(2n,n) gives A385413.
Cf. A214753, A215086.

b:= proc(n, k, l) option remember; `if`(n=0, 1,
       b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
H:= (n, k)-> A(n,k) -`if`(k=0, 0, A(n, k-1)):
T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, k_, L_] := b[n, k, L] = If[n == 0, 1, b[n - 1, k, Append[L, {1}]] + Sum[If[i == 1 || Length[L[[i]]] < Length[L[[i - 1]]], b[n - 1, k, ReplacePart[L, i -> Append[L[[i]], 1]]], 0] + Sum[If[L[[i, j]] < k && (i == 1 || L[[i, j]] < L[[i - 1, j]]) && (j == 1 || L[[i, j]] < L[[i, j - 1]]), b[n - 1, k, ReplacePart[L, i -> ReplacePart[L[[i]], j -> L[[i, j]] + 1]]], 0], {j, 1, Length[L[[i]]]}], {i, 1, Length[L]}]];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Min[n, k], {}]];
H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
T[n_, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k

A215087 Number of solid standard Young tableaux of n cells and height <= 2.

Original entry on oeis.org

1, 1, 3, 8, 26, 92, 372, 1566, 7086, 33550, 167504, 873226, 4764614, 26947632, 157926628, 954523378, 5945067490, 38060781922, 250345198424, 1688978186742, 11679437620552, 82652840640478, 598018846154666, 4418072084681592, 33298670603875846, 255782905412464810

Offset: 0

Alois P. Heinz, Aug 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..40
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=2 of A215086.

b:= proc(n, l) option remember; `if`(n=0, 1,
       b(n-1, [l[], [1]])+ add(`if`(i=1 or nops(l[i]) b(n, []):
seq(a(n), n=0..20);

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, Append[l, {1}]] + Sum[If[i == 1 || Length[l[[i]]] < Length[l[[i - 1]]], b[n - 1, ReplacePart[l, i -> Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]] < 2 && (i == 1 || l[[i, j]] < l[[i - 1, j]]) && (j == 1 || l[[i, j]] < l[[i, j - 1]]), b[n - 1, ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] + 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, Length[l]}]];
a[n_] := b[n, {}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 01 2023, after Alois P. Heinz *)

A320180 Number of solid standard Young tableaux of n cells and height <= 3.

Original entry on oeis.org

1, 1, 3, 9, 32, 126, 564, 2700, 13802, 74422, 422812, 2519962, 15731562, 102390662, 692779852, 4851392308, 35068814616, 261013287116, 1997276928946, 15689964298430, 126416353430344, 1043615269337354, 8819704403143576, 76229300376178238, 673190822341507644

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=3 of A215086.

A320181 Number of solid standard Young tableaux of n cells and height <= 4.

Original entry on oeis.org

1, 1, 3, 9, 33, 134, 622, 3106, 16720, 95140, 572638, 3621722, 24046746, 166799374, 1205714772, 9043761626, 70185282194, 561924753756, 4632640872604, 39262210040594, 341678806496748, 3050007040331626, 27902744129363050, 261376205797957108, 2504865765980461010

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=4 of A215086.

A320182 Number of solid standard Young tableaux of n cells and height <= 5.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 632, 3194, 17450, 101258, 622694, 4034892, 27489676, 196054798, 1459552844, 11297662922, 90648769970, 751732064930, 6429192275568, 56603447925754, 512309058662574, 4761135417168102, 45389581888641760, 443470604659923172, 4436647328514208908

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=5 of A215086.

A320183 Number of solid standard Young tableaux of n cells and height <= 6.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 633, 3206, 17574, 102444, 634004, 4139104, 28450444, 204936318, 1542796630, 12090425696, 98341974504, 827789394188, 7194985058192, 64450636832800, 594124150130062, 5628903025067656, 54753047168362054, 546261960780208350, 5584759231202910230

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=6 of A215086.

A320184 Number of solid standard Young tableaux of n cells and height <= 7.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 633, 3207, 17588, 102610, 635800, 4158270, 28646346, 206925540, 1562928436, 12295935458, 100464068006, 850030211990, 7431638110124, 67007477760634, 622164698873152, 5941062844925200, 58280974293944802, 586748859836258212, 6056670347690677380

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=7 of A215086.

A320185 Number of solid standard Young tableaux of n cells and height <= 8.

Original entry on oeis.org

1, 1, 3, 9, 33, 135, 633, 3207, 17589, 102626, 636014, 4160852, 28676800, 207267046, 1566709436, 12337438198, 100921477976, 855109095248, 7488671736398, 67655656640892, 629623562927598, 6027970886933100, 59306595225701588, 599011729235408510, 6205282969563030610

Offset: 0

Alois P. Heinz, Oct 07 2018

nonn

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Column k=8 of A215086.

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cp's OEIS Frontend

A214753 Number T(n,k) of solid standard Young tableaux of n cells and height = k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A207542 Number of solid standard Young tableaux with n cells.

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A215120 Number T(n,k) of solid standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A215087 Number of solid standard Young tableaux of n cells and height <= 2.

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A320180 Number of solid standard Young tableaux of n cells and height <= 3.

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A320181 Number of solid standard Young tableaux of n cells and height <= 4.

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A320182 Number of solid standard Young tableaux of n cells and height <= 5.

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A320183 Number of solid standard Young tableaux of n cells and height <= 6.

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A320184 Number of solid standard Young tableaux of n cells and height <= 7.

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A320185 Number of solid standard Young tableaux of n cells and height <= 8.

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