OFFSET
0,3
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
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 *)
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Matthew C. Russell, Feb 24 2012
STATUS
approved