Your data matches 59 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000987
(load all 30 compositions to match this statistic)
Mapping
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => 0
([(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(1,4),(2,3)],5)
 => 2
([(1,4),(2,3),(3,4)],5)
 => 3
([(0,1),(2,4),(3,4)],5)
 => 3
([(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 4
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000377
Mapping
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2
([],4)
 => [1,1,1,1]
 => [3,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [4]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => [3,2]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,2,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
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
(load all 2 compositions to match this statistic)
Mapping
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading 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]
 => [[1,3],[2]]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 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
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2
([],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
([(1,3),(2,3)],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)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(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),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([],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
([(2,4),(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
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 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
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,1),(2,4),(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
([(2,3),(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
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,1),(2,3),(2,4),(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
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
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
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
([],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
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
([],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
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
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
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
([],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
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
([],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
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
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
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => []
 => 0
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => []
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
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
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 2
([],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
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 3
([],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
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 4
Description
The number of ascents of a permutation.
Matching statistic: St000293
Mapping
Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
 => [3]
 => 1000 => 0100 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 0100 => 2
([],4)
 => [1,1,1,1]
 => 11110 => 01111 => 0
([(2,3)],4)
 => [2,1,1]
 => 10110 => 01011 => 1
([(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(0,3),(1,3),(2,3)],4)
 => [4]
 => 10000 => 01000 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 0110 => 2
([(0,3),(1,2),(2,3)],4)
 => [4]
 => 10000 => 01000 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 01001 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 01000 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 10000 => 01000 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 01000 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 01000 => 3
([],5)
 => [1,1,1,1,1]
 => 111110 => 011111 => 0
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010111 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 01101 => 2
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 010011 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 01010 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 010001 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => 100000 => 010000 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => 100000 => 010000 => 4
Description
The number of inversions of a binary word.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000026The position of the first return of a Dyck path. St000147The largest part of an integer partition. St000738The first entry in the last row of a standard tableau. St000839The largest opener of a set partition. St001389The number of partitions of the same length below the given integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000171The degree of the graph. St000369The dinv deficit of a Dyck path. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001645The pebbling number of a connected graph. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001725The harmonious chromatic number of a graph. St001330The hat guessing number of a graph. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000840The number of closers smaller than the largest opener in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001812The biclique partition number of a graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000080The rank of the poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001668The number of points of the poset minus the width of the poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation.