Filling -- a Young tableau stored column-wise

A Filling is a Young tableau of partition shape, stored as a list of its columns. A filling of shape $\mu = (\mu_1 \ge \mu_2 \ge \cdots \ge \mu_r)$ with entries from an alphabet $\{0, 1, \ldots, d-1\}$ is represented by new Filling from {c_1, c_2, ..., c_{mu_1}}, where $c_j$ is the list of entries in the $j$-th column read from top to bottom. The length of $c_j$ equals $\mu'_j$, the length of the $j$-th column (equivalently, the $j$-th part of the conjugate partition $\mu'$).

Because Schur modules are built from $\wedge^{\mu'_1}(E) \otimes \cdots \otimes \wedge^{\mu'_k}(E)$, the column-wise storage makes each column directly correspond to one wedge-power tensor factor. This is the dual convention to WeylFilling, which stores rows.

The tableau pretty-prints via its net method:

Conjugation of a Filling produces the tableau of conjugate (transposed) shape:

A filling is standard (in this package; equivalently, semistandard in the standard combinatorial sense) if its entries strictly increase down each column and weakly increase along each row. Over $\mathbb{Z}$ the standard fillings of shape $\mu$ with entries in $\{0,\ldots,d-1\}$ index a free basis of $S_\mu(R^d)$.