Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000770
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00077: Semistandard tableaux shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000770: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2]
=> 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,2]
=> [2]
=> 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2]
=> 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,2]
=> [2]
=> 2
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [2,2]
=> [2]
=> 2
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [4,2]
=> [2]
=> 2
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,2]
=> [2]
=> 2
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [4,2]
=> [2]
=> 2
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [3,3]
=> [3]
=> 3
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 2
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 4
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 4
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> 4
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [2,2]
=> [2]
=> 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [2,2]
=> [2]
=> 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [2,2]
=> [2]
=> 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [2,2]
=> [2]
=> 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [2,2]
=> [2]
=> 2
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St001904
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00305: Permutations parking functionParking functions
St001904: Parking functions ⟶ ℤResult quality: 33% values known / values provided: 35%distinct values known / distinct values provided: 33%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 1
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 1
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 4
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 4
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 4
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 2
[[2,2,0,0],[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[2,1,1,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[2,1,1,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[2,1,1,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[2,1,1,0],[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[[2,1,1,0],[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 1
[[5,2],[2]]
=> [[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => ? = 2
[[5,2],[3]]
=> [[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [4,5,1,2,3,6,7] => ? = 2
[[5,2],[4]]
=> [[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[5,2],[5]]
=> [[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 2
[[4,3],[3]]
=> [[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3
[[4,3],[4]]
=> [[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 3
[[4,2,0],[2,0],[0]]
=> [[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2,0],[2,0],[1]]
=> [[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2,0],[2,0],[2]]
=> [[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2,0],[2,1],[1]]
=> [[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2
[[4,2,0],[2,1],[2]]
=> [[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2,0],[2,2],[2]]
=> [[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2,0],[3,0],[0]]
=> [[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2,0],[3,0],[1]]
=> [[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2,0],[3,0],[2]]
=> [[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2,0],[3,0],[3]]
=> [[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2,0],[3,1],[1]]
=> [[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [2,5,1,3,4,6] => ? = 2
[[4,2,0],[3,1],[2]]
=> [[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [3,5,1,2,4,6] => ? = 2
[[4,2,0],[3,1],[3]]
=> [[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2,0],[3,2],[2]]
=> [[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
Description
The length of the initial strictly increasing segment of a parking function.
Matching statistic: St001882
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001882: Signed permutations ⟶ ℤResult quality: 33% values known / values provided: 35%distinct values known / distinct values provided: 33%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2 - 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3 - 1
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 4 - 1
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 4 - 1
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 4 - 1
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,2,0,0],[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,1,1,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1,0],[2,1,0],[2,1],[2]]
=> [[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,1,1,0],[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[2,1,1,0],[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,1,1,0],[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[5,2],[2]]
=> [[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => ? = 2 - 1
[[5,2],[3]]
=> [[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [4,5,1,2,3,6,7] => ? = 2 - 1
[[5,2],[4]]
=> [[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2 - 1
[[5,2],[5]]
=> [[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 2 - 1
[[4,3],[3]]
=> [[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3 - 1
[[4,3],[4]]
=> [[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 3 - 1
[[4,2,0],[2,0],[0]]
=> [[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2,0],[2,0],[1]]
=> [[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2,0],[2,0],[2]]
=> [[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2,0],[2,1],[1]]
=> [[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2 - 1
[[4,2,0],[2,1],[2]]
=> [[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2,0],[2,2],[2]]
=> [[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
[[4,2,0],[3,0],[0]]
=> [[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2,0],[3,0],[1]]
=> [[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2,0],[3,0],[2]]
=> [[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2,0],[3,0],[3]]
=> [[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2,0],[3,1],[1]]
=> [[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [2,5,1,3,4,6] => ? = 2 - 1
[[4,2,0],[3,1],[2]]
=> [[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [3,5,1,2,4,6] => ? = 2 - 1
[[4,2,0],[3,1],[3]]
=> [[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 - 1
[[4,2,0],[3,2],[2]]
=> [[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 - 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001857
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001857: Signed permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 17%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ? = 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => ? = 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => ? = 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 2
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 1
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 1
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 4
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 4
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 4
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[5],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[5],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[5],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
Description
The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001926
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001926: Signed permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 17%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2 + 2
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ? = 1 + 2
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => ? = 1 + 2
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => ? = 1 + 2
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2 + 2
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2 + 2
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2 + 2
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3 + 2
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 + 2
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 + 2
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 + 2
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 + 2
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 + 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 + 2
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 2
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 + 2
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 + 2
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 + 2
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 + 2
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 4 + 2
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ? = 4 + 2
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ? = 4 + 2
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2 + 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2 + 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => ? = 2 + 2
[[2,2,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => ? = 2 + 2
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[5],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[5],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[5],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[5],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[6]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[5]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 1 + 2
Description
Sparre Andersen's position of the maximum of a signed permutation. For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|})$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Andersen's statistic does not matter so long as no sums or differences of any subset of the $c_i$'s is zero. The choice of powers of $2$ is just a convenient choice. This returns the largest position of the maximum value in the $x$-tuple. This is related to the ''discrete arcsine distribution''. The number of signed permutations with value equal to $k$ is given by $\binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}$. This statistic is equidistributed with Sparre Andersen's `Number of Positives' statistic.
Matching statistic: St001754
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00208: Permutations lattice of intervalsLattices
St001754: Lattices ⟶ ℤResult quality: 4% values known / values provided: 4%distinct values known / distinct values provided: 17%
Values
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 2
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? = 2
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? = 2
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? = 3
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 2
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2
[[3,2,0],[3,1],[2]]
=> [[1,1,2],[2,3]]
=> [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2
[[3,2,0],[3,1],[3]]
=> [[1,1,1],[2,3]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,2,0],[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
[[3,2,0],[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[3,1,1],[1,1],[1]]
=> [[1,3,3],[2],[3]]
=> [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[[3,1,1],[2,1],[1]]
=> [[1,2,3],[2],[3]]
=> [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[3,1,1],[2,1],[2]]
=> [[1,1,3],[2],[3]]
=> [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 1
[[3,1,1],[3,1],[1]]
=> [[1,2,2],[2],[3]]
=> [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 1
[[3,1,1],[3,1],[2]]
=> [[1,1,2],[2],[3]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[3,1,1],[3,1],[3]]
=> [[1,1,1],[2],[3]]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[[2,2,1],[2,1],[1]]
=> [[1,2],[2,3],[3]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[[2,2,1],[2,1],[2]]
=> [[1,1],[2,3],[3]]
=> [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 4
[[2,2,1],[2,2],[2]]
=> [[1,1],[2,2],[3]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 4
[[2,2,0,0],[2,0,0],[0,0],[0]]
=> [[3,3],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,0,0],[2,0],[0]]
=> [[2,2],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,0,0],[2,0],[2]]
=> [[1,1],[4,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 2
[[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0],[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,0,0],[0,0],[0]]
=> [[3,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,0,0],[1,0],[0]]
=> [[2,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,1,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,5]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,2,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,2,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,2,0,0],[2,1,0],[2,1],[1]]
=> [[1,2],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[2,2,0,0,0],[2,2,0,0],[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
Description
The number of tolerances of a finite lattice. Let $L$ be a lattice. A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join. Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$. The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].