Identifier
Values
[1,0] => [(1,2)] => [2,1] => [[1],[2]] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [[1,3],[2,4]] => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => [[1,4],[2],[3]] => 3
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [[1,3,5],[2,4,6]] => 3
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [[1,3,6],[2,4],[5]] => 4
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [[1,4,5],[2,6],[3]] => 4
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [[1,4,6],[2,5],[3]] => 4
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [[1,5,6],[2],[3],[4]] => 6
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [[1,3,5,7],[2,4,6,8]] => 4
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [[1,3,5,8],[2,4,6],[7]] => 5
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [[1,3,6,7],[2,4,8],[5]] => 5
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [[1,3,6,8],[2,4,7],[5]] => 5
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [[1,3,7,8],[2,4],[5],[6]] => 7
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [[1,4,5,7],[2,6,8],[3]] => 5
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [[1,4,5,8],[2,6],[3,7]] => 6
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [[1,4,6,7],[2,5,8],[3]] => 5
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [[1,4,6,8],[2,5,7],[3]] => 5
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [[1,4,7,8],[2,5],[3],[6]] => 7
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [[1,5,6,7],[2,8],[3],[4]] => 7
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [[1,5,6,8],[2,7],[3],[4]] => 7
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [[1,5,7,8],[2,6],[3],[4]] => 7
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [[1,6,7,8],[2],[3],[4],[5]] => 10
[1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [[1,3,5,7,9],[2,4,6,8,10]] => 5
[1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => [3,5,2,7,4,9,6,10,8,1] => [[1,4,6,8,10],[2,5,7,9],[3]] => 6
[1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => [4,6,8,3,9,5,10,7,2,1] => [[1,5,7,9,10],[2,6,8],[3],[4]] => 8
[1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => [5,7,8,9,4,10,6,3,2,1] => [[1,6,8,9,10],[2,7],[3],[4],[5]] => 11
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => 15
[1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 6
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Description
The leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Map
Robinson-Schensted insertion tableau
Description
Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.