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Matching statistic: St000377
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%
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
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
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
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 15
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: St000228
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: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
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
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 15
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●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
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> 1000100010 => ? => ? = 9
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? => ? = 6
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> 100011011110 => ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> [6,2,1,1,1]
=> 10000101110 => 01000010111 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> [6,2,2,1,1]
=> 10000110110 => ? => ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> [6,2,1,1,1]
=> 10000101110 => 01000010111 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> [6,2,1,1,1]
=> 10000101110 => 01000010111 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> [6,3,1,1,1]
=> 10001001110 => ? => ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> [6,2,1,1,1]
=> 10000101110 => 01000010111 => ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> [6,2,1,1,1]
=> 10000101110 => 01000010111 => ? = 6
Description
The number of inversions of a binary word.
Matching statistic: St000394
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 10
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 15
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 7
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 9
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 6
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 6
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> [6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> [6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
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
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ? = 15 + 1
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? = 9 + 1
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 1
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 9 + 1
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 + 1
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 9 + 1
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 + 1
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 1
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 1
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 1
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 1
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 1
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? = 5 + 1
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5 + 1
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5 + 1
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5 + 1
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 5 + 1
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? = 5 + 1
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ? = 6 + 1
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ? = 6 + 1
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ? = 6 + 1
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ? = 6 + 1
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ? = 6 + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> [6,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11],[12]]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> [6,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6 + 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: St000738
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? = 15 + 1
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> [[1,5,12],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 9 + 1
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 8 + 1
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 9 + 1
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 9 + 1
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 9 + 1
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 9 + 1
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 8 + 1
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 8 + 1
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 10 + 1
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? = 5 + 1
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 5 + 1
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 5 + 1
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 5 + 1
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? = 5 + 1
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? = 5 + 1
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [[1,5,6,10,11,12,13],[2,8,9],[3],[4],[7]]
=> ? = 6 + 1
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> [[1,4,9,10,11,12,13],[2,6],[3,8],[5],[7]]
=> ? = 6 + 1
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [[1,5,6,10,11,12,13],[2,8,9],[3],[4],[7]]
=> ? = 6 + 1
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [[1,5,6,10,11,12,13],[2,8,9],[3],[4],[7]]
=> ? = 6 + 1
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> [[1,4,9,10,11,12,13],[2,6],[3,8],[5],[7]]
=> ? = 6 + 1
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 8 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> [6,2,2,1,1]
=> [5,3,1,1,1,1]
=> [[1,6,7,11,12],[2,9,10],[3],[4],[5],[8]]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> [6,3,1,1,1]
=> [5,2,2,1,1,1]
=> [[1,5,10,11,12],[2,7],[3,9],[4],[6],[8]]
=> ? = 7 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000074
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: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
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
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[5,3,3,1,0,0,0,0,0,0,0,0],[5,3,3,0,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[5,4,3,2,1,0,0,0,0,0,0,0,0,0,0],[5,4,3,2,0,0,0,0,0,0,0,0,0,0],[5,4,3,1,0,0,0,0,0,0,0,0,0],[5,4,3,0,0,0,0,0,0,0,0,0],[5,4,2,0,0,0,0,0,0,0,0],[5,4,1,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ?
=> ? = 15
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ?
=> ? = 9
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ?
=> ? = 9
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ?
=> ? = 9
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ?
=> ? = 9
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ?
=> ? = 9
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[5,3,3,1,0,0,0,0,0,0,0,0],[5,3,3,0,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ?
=> ? = 8
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)
(load all 2 compositions to match this statistic)
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: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
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
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,6,9,12],[2,7,10],[3,8,11],[4],[5]]
=> ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,6,10,13,15],[2,7,11,14],[3,8,12],[4,9],[5]]
=> ? = 10
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> [[1,7,12,16,19,21],[2,8,13,17,20],[3,9,14,18],[4,10,15],[5,11],[6]]
=> ? = 15
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ?
=> ? = 9
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ?
=> ? = 9
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> [[1,9],[2,10],[3,11],[4],[5],[6],[7],[8]]
=> ? = 9
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ?
=> ? = 9
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> [[1,9],[2,10],[3,11],[4],[5],[6],[7],[8]]
=> ? = 9
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> [[1,6,8,9,10,11],[2,7],[3],[4],[5]]
=> ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> [[1,6,8,9,10,11],[2,7],[3],[4],[5]]
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> [[1,6,8,9,10,11],[2,7],[3],[4],[5]]
=> ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12],[13]]
=> ?
=> ? = 6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,6,9,12],[2,7,10],[3,8,11],[4],[5]]
=> ? = 8
([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
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: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
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
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11,12] => ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? = 10
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? = 15
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 8
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ? => ? = 9
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 8
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 9
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 9
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 9
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 9
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 8
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 5
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,7,8,12,13],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 6
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 6
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [[1,6,11,12,13],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> [[1,7,8,12,13],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11,12] => ? = 8
([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
Description
The number of ascents of a permutation.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000369The dinv deficit of a Dyck path. St000502The number of successions of a set partitions. St000211The rank of the set partition. St000093The cardinality of a maximal independent set of vertices of a graph. 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. St000362The size of a minimal vertex cover of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001668The number of points of the poset minus the width of the poset. St001626The number of maximal proper sublattices of a lattice.
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