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A207542
Number of solid standard Young tableaux with n cells.
4
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
0,3
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.
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
Rows sums of A214753.
Main diagonal of A215086.
Column k=0 of A215120. - Alois P. Heinz, May 12 2014
Sequence in context: A320185 A320186 A320187 * A009212 A153344 A193110
KEYWORD
nonn,hard
AUTHOR
Matthew C. Russell, Feb 24 2012
STATUS
approved