Your data matches 33 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001668
(load all 2 compositions to match this statistic)
Mapping
St001668: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => 1
([(0,2),(2,1)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,2),(1,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => 1
([(0,2),(0,3),(3,1)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => 2
([],5)
 => 0
([(3,4)],5)
 => 1
([(2,3),(2,4)],5)
 => 1
([(1,2),(1,3),(1,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(4,2)],5)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(2,3),(3,4)],5)
 => 2
([(1,4),(4,2),(4,3)],5)
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => 2
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000362
(load all 3 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
([],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
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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
([],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
([(0,4),(1,3),(1,4),(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.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. 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. St000459The hook length of the base cell of a partition. St000502The number of successions of a set partitions. St000672The number of minimal elements in Bruhat order not less than the permutation. St000728The dimension of a set partition. St001298The number of repeated entries in the Lehmer code of a permutation. St001480The number of simple summands of the module J^2/J^3. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000240The number of indices that are not small excedances. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001626The number of maximal proper sublattices of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001330The hat guessing number of a graph.