Identifier
Values
[(1,2)] => [2,1] => [1,1,0,0] => [2,3,1] => 1
[(1,2),(3,4)] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[(1,3),(2,4)] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 2
[(1,4),(2,3)] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 2
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 3
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[(1,8),(2,3),(4,5),(6,7)] => [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,5),(6,7)] => [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,4),(6,7)] => [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,4),(5,7)] => [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,4),(5,6)] => [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,5),(4,6)] => [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,5),(4,7)] => [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,6),(4,7)] => [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,6),(5,7)] => [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,6),(5,7)] => [8,3,2,6,7,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,7),(5,6)] => [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,7),(5,6)] => [8,4,7,2,6,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,7),(4,6)] => [8,5,7,6,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,7),(4,5)] => [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,6),(4,5)] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
[(1,10),(2,3),(4,5),(6,7),(8,9)] => [10,3,2,5,4,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,7),(8,9)] => [10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,4),(5,6),(8,9)] => [10,7,4,3,6,5,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,6),(7,8)] => [10,9,4,3,6,5,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,7),(5,6),(8,9)] => [10,3,2,7,6,5,4,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,6),(4,5),(8,9)] => [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,6),(4,5),(7,8)] => [10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,6),(7,8)] => [10,3,2,9,6,5,8,7,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)] => [10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)] => [10,8,9,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,5),(6,9),(7,8)] => [10,3,2,5,4,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,9),(7,8)] => [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,8),(6,7)] => [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,8),(6,7)] => [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.