A Young diagram, or Ferrers diagram, is a collection of boxes, or cells, arranged in left--justified rows, with a weakly decreasing number of boxes in each row. Listing the number of boxes in each row gives a partition (its shape) of n, where n is the total number of boxes in the diagram. Conversely, each partition corresponds to a unique Young diagram. A filling of a Young diagram places a positive integer in each box. A Young tableau, or simply tableau, is a filling which has i) weakly increasing entries across each row, and ii) strictly increasing entries down each column. Flipping a diagram over its main diagonal gives the Conjugate diagram, its shape being the ConjugatePartition of the original shape.
A Skew diagram or Skew Shape is the diagram obtained by removing a smaller Young diagram from a larger on that contains it. A Skew tableau is a filling on a skew diagram obeying the same restrictions on entries.
Every tableau has an outer shape and a skew shape, both partitions. Calculus of the tableau takes place by way of standard row insertion and Jeu De Taquin algorithms.
s is a sequence of sequences of positive integers which are the entries of the tableau. To form a skew tableau, zeros are to be used to denote skew entries.
Returns a standard skew tableau of outer shape P1 and skew shape P2, entries are from biggest to smallest right to left bottom to top.
Returns a (non--skew) standard tableau of outer shape P1, entries are from biggest to smallest right to left and bottom to top.
Each word corresponds to a unique tableau. This tableau is obtained through row insertion of w (from the left) into the empty tableau.
Each Partition corresponds to a Young diagram, and each position on that diagram has an associated hook. The hook consists of all boxes in its row to the right of the box, and all boxes in its column below the hook. HookLength returns the length of the hook of the (i, j)th position of the Young Diagram for the partition P.
Returns the number of standard tableau constructible on the shape described by the partition P.
Returns the number of tableau constructible on the shape described by the partition P, and with entries coming from [m].
The DiagonalSum of two tableau is formed by first having a rectangle of empty squares, with the same number of columns as t1 and the same number of rows as t2. Then t1 is attached below the rectangle and t2 attached to right of it.
If t is skew then it will be rectified using Jeu de Taquin to remove any skewed entries.
The value i is row inserted into tableau t using the standard row insertion algorithm.
The word w is row inserted left to right into the tableau t using the standard row insertion algorithm.
The content of a tableau is a sequence where the ith position denotes the number of occurrences of i in the tableau.
Returns the row word of the tableau, reading entries left to right, bottom to top.
Returns the column word of tableau t, reading the entries from bottom to top, left to right.
Returns the partition denoting the outer shape of the tableau.
Returns the partition (with trailing zeros) of the skew shape of the tableau.
Returns the weight of the tableau, which is the number of (non--skew) entries
Returns the skew weight of the tableau, which is the number of skew entries
The number of rows in the tableau.
Returns the number of skewed rows in the tableau.
Returns the ith row of tableau t. If tableau is skew, then skewed positions will be filled with zeros
Returns an sequence of sequence's which are the rows of the tableau (skewed positions filled with zeros)
Returns the index of the first non--skew entry of the ith row. If the row has no non--skew entries then the index will be 1 greater than the length of the row, ie out of bounds.
Returns the index of the last entry of the tableau in the ith row, which is the length of the ith row.
Returns the index of the first non--skew entry of the jth column. If the column has no non--skew entries then the index will be 1 greater than the length of the column, ie out of bounds.
Returns the index of the last entry of the tableau in the jth column, which is the length of the jth column.
Returns the length of the skewed portion of the ith row (zero if no skewed portion)
Returns the length of the skewed portion of the jth column (zero if no skewed portion)
Returns true if t is a standard tableau. ie has entries 1 ... n, where n is the weight of the tableau. No repeats are allowed.
Returns true if t is a skew tableau.
Returns true if t is a Littlewood--Richardson Skew tableau, i.e. if the content of t forms a reverse lattice word.[Next][Prev] [Right] [Left] [Up] [Index] [Root]