Processing math: 62%

Your data matches 77 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000029: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The depth of a permutation. This is given by dp(σ)=σi>i(σii)=|{ij:σi>j}|. The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] i|σii|. Permutations with depth at most 1 are called ''almost-increasing'' in [5].
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000030: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The sum of the descent differences of a permutations. This statistic is given by πiDes(π)(πiπi+1). See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000057
Mp00080: Set partitions to permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00155: Standard tableaux promotionStandard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [[1]]
=> [[1]]
=> 0
{{1,2}}
=> [2,1] => [[1],[2]]
=> [[1],[2]]
=> 0
{{1},{2}}
=> [1,2] => [[1,2]]
=> [[1,2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [[1,2],[3]]
=> [[1,3],[2]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> 0
{{1},{2,3}}
=> [1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [[1,2,4],[3]]
=> [[1,2,3],[4]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [[1,2],[3],[4]]
=> [[1,3],[2],[4]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [[1,3],[2],[4]]
=> [[1,4],[2],[3]]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,3],[4]]
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [[1,3],[2],[4]]
=> [[1,4],[2],[3]]
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,3],[2],[4]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [[1,3,4],[2],[5]]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> [[1,3,4],[2,5]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> [[1,3],[2],[4],[5]]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> [[1,4,5],[2],[3]]
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[1,2],[3,5],[4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> [[1,3,4],[2],[5]]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> [[1,2],[3,5],[4]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> [[1,5],[2],[3],[4]]
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> 3
Description
The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000394
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 0
{{1,2}}
=> [2,1] => [1,2] => [1,0,1,0]
=> 0
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000463
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00066: Permutations inversePermutations
St000463: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
Description
The number of admissible inversions of a permutation. Let w=w1,w2,,wk be a word of length k with distinct letters from [n]. An admissible inversion of w is a pair (wi,wj) such that 1i<jk and wi>wj that satisfies either of the following conditions: 1<i and wi1<wi or there is some l such that i<l<j and wi<wl.
Matching statistic: St001726
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
St001726: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,5,2,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,5,2,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,5,4,3,2] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,3,2,5] => 2
Description
The number of visible inversions of a permutation. A visible inversion of a permutation π is a pair i<j such that π(j)min.
Matching statistic: St001861
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001861: Signed permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
Description
The number of Bruhat lower covers of a permutation. This is, for a signed permutation \pi, the number of signed permutations \tau having a reduced word which is obtained by deleting a letter from a reduced word from \pi.
Matching statistic: St001869
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
St001869: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
Description
The maximum cut size of a graph. A '''cut''' is a set of edges which connect different sides of a vertex partition V = A \sqcup B.
Matching statistic: St001894
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001894: Signed permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,3,4,2] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
Description
The depth of a signed permutation. The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
St000572: Set partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 2
{{1,4},{2,3,5}}
=> 3
{{1,4},{2,3},{5}}
=> 2
Description
The dimension exponent of a set partition. This is \sum_{B\in\pi} (\max(B) - \min(B) + 1) - n where the summation runs over the blocks of the set partition \pi of \{1,\dots,n\}. It is thus equal to the difference [[St000728]] - [[St000211]]. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
The following 67 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000957The number of Bruhat lower covers of a permutation. St000491The number of inversions of a set partition. St000728The dimension of a set partition. St000225Difference between largest and smallest parts in a partition. St001596The number of two-by-two squares inside a skew partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001964The interval resolution global dimension of a poset. St001867The number of alignments of type EN of a signed permutation. St001438The number of missing boxes of a skew partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001864The number of excedances of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation.