Your data matches 91 different statistics following compositions of up to 3 maps.
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Matching statistic: St000362
(load all 5 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => 0
([],2)
 => ([(0,1)],2)
 => ([],2)
 => 0
([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 0
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 0
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 0
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
Description
The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000377
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 0
([],2)
 => [1,1]
 => [2]
 => 0
([(0,1)],2)
 => [2]
 => [1,1]
 => 1
([],3)
 => [1,1,1]
 => [2,1]
 => 0
([(1,2)],3)
 => [2,1]
 => [3]
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [3]
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => 1
([],4)
 => [1,1,1,1]
 => [3,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [4]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [4]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([],5)
 => [1,1,1,1,1]
 => [3,2]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [5]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [5]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [5]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [5]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [2,2,1]
 => 2
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 0
([],2)
 => [1,1]
 => [2]
 => 0
([(0,1)],2)
 => [2]
 => [1,1]
 => 1
([],3)
 => [1,1,1]
 => [3]
 => 0
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => 1
([],4)
 => [1,1,1,1]
 => [4]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([],5)
 => [1,1,1,1,1]
 => [5]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => 2
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: 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)
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000074
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => [[1]]
 => 0
([],2)
 => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 0
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 0
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
Description
The number of special entries. An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => [[1]]
 => 0
([],2)
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => {{1}}
 => 0
([],2)
 => [1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 0
([(0,1)],2)
 => [2]
 => [[1,2]]
 => {{1,2}}
 => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 2
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000228
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => []
 => 0
([],2)
 => [1,1]
 => [2]
 => []
 => 0
([(0,1)],2)
 => [2]
 => [1,1]
 => [1]
 => 1
([],3)
 => [1,1,1]
 => [3]
 => []
 => 0
([(1,2)],3)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([],4)
 => [1,1,1,1]
 => [4]
 => []
 => 0
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([],5)
 => [1,1,1,1,1]
 => [5]
 => []
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => [1] => 0
([],2)
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [1,2] => 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
([(0,1),(0,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 2
([(0,2),(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(1,2),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000293
Mapping
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 10 => 01 => 0
([],2)
 => [1,1]
 => 110 => 011 => 0
([(0,1)],2)
 => [2]
 => 100 => 010 => 1
([],3)
 => [1,1,1]
 => 1110 => 0111 => 0
([(1,2)],3)
 => [2,1]
 => 1010 => 0101 => 1
([(0,1),(0,2)],3)
 => [2,1]
 => 1010 => 0101 => 1
([(0,2),(2,1)],3)
 => [3]
 => 1000 => 0100 => 2
([(0,2),(1,2)],3)
 => [2,1]
 => 1010 => 0101 => 1
([],4)
 => [1,1,1,1]
 => 11110 => 01111 => 0
([(2,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(1,2),(1,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(0,1),(0,2),(0,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 0110 => 2
([(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 0110 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 0110 => 2
([(0,3),(2,1),(3,2)],4)
 => [4]
 => 10000 => 01000 => 3
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => 10010 => 01001 => 2
([],5)
 => [1,1,1,1,1]
 => 111110 => 011111 => 0
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(2,3),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(0,4),(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 01101 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,1]
 => 11010 => 01101 => 2
Description
The number of inversions of a binary word.
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000369The dinv deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000738The first entry in the last row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000741The Colin de Verdière graph invariant. St000144The pyramid weight of the Dyck path. St001668The number of points of the poset minus the width of the poset. St000454The largest eigenvalue of a graph if it is integral. 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. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001480The number of simple summands of the module J^2/J^3. St000264The girth of a graph, which is not a tree. St001720The minimal length of a chain of small intervals in a lattice. St000145The Dyson rank of a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001626The number of maximal proper sublattices of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000171The degree of the graph. St000387The matching number of a graph. St000552The number of cut vertices of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001393The induced matching number of a graph. St001689The number of celebrities in a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000537The cutwidth of a graph. St000553The number of blocks of a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000469The distinguishing number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph.