A207542 Number of solid standard Young tableaux with n cells.

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

Author

Matthew C. Russell, Feb 24 2012

Keywords

• nonn
• hard

Comments

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

Links

• 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

Crossrefs

Rows sums of A214753.

Main diagonal of A215086.

Column k=0 of A215120. - Alois P. Heinz, May 12 2014

Programs

Mathematica

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 *)