Identifier
Values
[[1]] => [[1]] => [[1]] => 0
[[1,2]] => [[1,2]] => [[2,0],[1]] => 0
[[1],[2]] => [[1,2]] => [[2,0],[1]] => 0
[[1,2,3]] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[[1,3],[2]] => [[1,2],[3]] => [[2,1,0],[2,0],[1]] => 0
[[1,2],[3]] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[[1],[2],[3]] => [[1,2],[3]] => [[2,1,0],[2,0],[1]] => 0
[[1,2,3,4]] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,3,4],[2]] => [[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => 2
[[1,2,4],[3]] => [[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => 1
[[1,2,3],[4]] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,3],[2,4]] => [[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => 2
[[1,2],[3,4]] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,4],[2],[3]] => [[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => 0
[[1,3],[2],[4]] => [[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => 2
[[1,2],[3],[4]] => [[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => 1
[[1],[2],[3],[4]] => [[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => 0
[[1,2,3,4,5]] => [[1,2,3,4,5]] => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 3
[[1,3,4,5],[2]] => [[1,2,4,5],[3]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 3
[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 3
[[1,2,3,5],[4]] => [[1,2,3,4],[5]] => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,2,3,4],[5]] => [[1,2,3,4,5]] => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 3
[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 4
[[1,2,5],[3,4]] => [[1,2,3,4],[5]] => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,3,4],[2,5]] => [[1,2,4,5],[3]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 3
[[1,2,4],[3,5]] => [[1,2,3,5],[4]] => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 3
[[1,2,3],[4,5]] => [[1,2,3,4,5]] => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 3
[[1,4,5],[2],[3]] => [[1,2,5],[3],[4]] => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => 3
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 2
[[1,2,5],[3],[4]] => [[1,2,3],[4],[5]] => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 1
[[1,3,4],[2],[5]] => [[1,2,4,5],[3]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 3
[[1,2,4],[3],[5]] => [[1,2,3,5],[4]] => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 3
[[1,2,3],[4],[5]] => [[1,2,3,4],[5]] => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,4],[2,5],[3]] => [[1,2,5],[3],[4]] => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => 3
[[1,3],[2,5],[4]] => [[1,2,4,5],[3]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 3
[[1,2],[3,5],[4]] => [[1,2,3,5],[4]] => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 3
[[1,3],[2,4],[5]] => [[1,2,4],[3,5]] => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 4
[[1,2],[3,4],[5]] => [[1,2,3,4],[5]] => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 2
[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]] => 0
[[1,4],[2],[3],[5]] => [[1,2,5],[3],[4]] => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => 3
[[1,3],[2],[4],[5]] => [[1,2,4],[3],[5]] => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => 2
[[1,2],[3],[4],[5]] => [[1,2,3],[4],[5]] => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => 1
[[1],[2],[3],[4],[5]] => [[1,2],[3],[4],[5]] => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]] => 0
[[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 4
[[1,2,3,4,5],[6]] => [[1,2,3,4,5,6]] => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 4
[[1,2,3,4],[5,6]] => [[1,2,3,4,5,6]] => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 4
[[1,2,3],[4,5,6]] => [[1,2,3,4,5,6]] => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => 4
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Description
Number of free entries.
The tiling of a pattern is the finest partition of the entries in
the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belong to exactly one tile.
A tile is free if it do not intersect any of the first and the last row.
This statistic is the total number of entries that belong to a free tile.
Map
catabolism
Description
Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).
Map
to Gelfand-Tsetlin pattern
Description
Sends a tableau to its corresponding Gelfand-Tsetlin pattern.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.