Identifier
Values
[1,0] => [(1,2)] => [2,1] => [1,2] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [1,4,2,3] => 2
[1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [1,4,2,3,6,5] => 2
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [1,6,2,3,4,5] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,6,2,3,4,5] => 2
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,2,5,3,4,6] => 2
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 2
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 2
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 2
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 2
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 2
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 2
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 1
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 2
[1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 2
[1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 2
[1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 2
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 1
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Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
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.