`a(n)` is the number of linear extensions of poset `P_n`. (Exercise J is an simpler example of counting linear extensions.)
One can figure out what the Hasse diagram of `P_n` is by observing Hasse diagrams of `P_2`, `P_3`, `P_4` and `P_5` in the picture.
Find `a(n)`.
Updated 04/13/2015 and remarked by Ran Pan
`a(n)` also counts the number of maximum packings of pattern `Q` in column-strict arrays of size `3\times (n+1)`.
`a(n)` also counts the number of standard Young tableaux of shape `n^3` (French notation) avoiding pattern `T`, i.e., the number of standard Young tableaux of shape `n^3` (French notation) such that for any element in the tableau, its upper element is larger than its right element.
Updated 02/26/2015 and contributed by Quang Tran Bach
`a(n)` counts sequence A181197 on OEIS.
The hook-length formula for shifted Young tableaux is used for this problem. Solution with details will be posted here soon.