The First , 30, terms of the Integer Sequence
Number of Solid Standard Young Tableaux with n cells
By Shalosh B. Ekhad, Rutgers University
(Email: ShaloshBEkahd at gmail dot com)
Suppose you have n people of all different heights
who are to take seats in a cubical 3D auditorium so as
to form a formation that is left-justified, front-justified
and bottom-justified, in other words they occupy a plane partition
with n cells. In addition, there are three circus acts
one on the left side, one on the front side, and one below
(it is a glass floor). You don't want any one blocking the view,
so if you look at any line from left to right, and
from front to back, and
from bottom to top, the heights of the people are increasing.
The enumerating sequence is:
[1, 3, 9, 33, 135, 633, 3207, 17589, 102627, 636033, 4161141, 28680717,
207318273, 1567344549, 12345147705, 101013795753, 856212871761,
7501911705747, 67815650852235, 631574151445665, 6051983918989833,
59605200185016639, 602764245172225251, 6252962956009863363,
66482211459036254169, 723810526382641418667, 8062440364611311185977,
91804267420894431624357, 1067720130017504052805449,
12673922788286515247094267]
On Feb. 8, 2012, this sequence did not yet exist in Sloane.
Let's hope that soon someone will fill this much needed gap.
Comment:
A google search (Feb. 8, 2012) revealed that the terms for
n=12,14,15,17 appear in the interesting article
"On the asymptotics of higher-dimensional partitions"
by Srivatsan Balakrishnan, Suresh Govindarajan
and Naveen S. Prabhakar, http://arxiv.org/abs/1105.6231
but there these numbers are not a goal for themselves, and
tangent to the main thread.
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