Convex Quadrilateral Test

I use standard facts about symmetric functions that can be found, e.g, in Chapter 7 of Enumerative Combinatorics, vol. 2. Let $s_\lambda$ denote a Schur function and $p_1=s_1=x_1+x_2+\cdots$. Then $\frac{\partial s_\lambda}{\partial p_1}=\sum_\mu s_\mu$, where the $\mu$'s are obtained by removing a single box from $\lambda$. Moreover, the coefficient of $q^n$ in $s_\lambda(q,q^2,q^3,\dots)$ is equal to the number of SSYT with entries summing to $n$. Thus if we let $c_n$ be the coefficient of $q^n$ in the symmetric function $f$ defined by $$ f+\frac{\partial}{\partial p_1}f = \sum_\lambda s_\lambda\cdot s_\lambda(q,q^2,q^3,\dots), $$ and if $c_n=\sum a_{\mu,n}s_\mu$, then we need $a_{\mu,n}$ copies of the shape $\mu$ in a set of fathers generating all SSYT with entries summing to $n$. Hence we need to show that $a_{\mu,n}\geq 0$. Now $$ \sum_\lambda s_\lambda\cdot s_\lambda(q,q^2,q^3,\dots) = \exp \sum_{n\geq 1} \frac{q^n}{1-q^n}p_n. $$ This leads to a simple linear first-order differential equation with solution $$ f = (1-q) \sum_\lambda s_\lambda\cdot s_\lambda(q,q^2,q^3,\dots). $$ If $h_u$ denotes the hook length of the square $u$ of $\lambda$, then $$ s_\lambda(q,q^2,q^3,\dots) = \frac{q^{b(\lambda)}}{\prod_{u\in \lambda} (1-q^{h_u})}, $$ where $b_\lambda=\sum i\lambda_i$. Since there is always a hook length equal to one, the power series $(1-q)s_\lambda(q,q^2,q^3,\dots)$ will be a product of factors of the form $1/(1-q^h)$, $h\geq 1$, so will have nonnegative coefficients as desired. Thus we have not just an existence proof, but a precise generating function for the number of fathers of each shape.

DATETIME only allows three places of millisecond precision. You'll either need to trim the trailing places beyond the first three milliseconds or if you're using any version of SQL Server 2008 or later you can use DATETIME2.

DATETIME2 allows up to seven places of millisecond precision and the precision level can be explicitly set for your table or query. Further details and differences can be found at datetime2 (Transact-SQL).

You can verify by executing the following CAST query:

SELECT CAST('2015-12-02 20:40:37.8130000' AS DATETIME2)

In this particular case, the issue was with an incorrectly configured memcache server settings, the Drupal environment in question was attempting to communicate with a memcache instance that had a firewall dropping all connections.

The fix was to configure memcache servers in the Drupal settings file. Hope this helps someone else.